Calculus Examples

$lim_{x \to 0} \frac{\sqrt{x + 1} - \sqrt{2 x + 1}}{\sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0} \sqrt{x + 1} - \sqrt{2 x + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \sqrt{x + 1} - lim_{x \to 0} \sqrt{2 x + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} x + 1} - lim_{x \to 0} \sqrt{2 x + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + lim_{x \to 0} 1} - lim_{x \to 0} \sqrt{2 x + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + 1} - lim_{x \to 0} \sqrt{2 x + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.5

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} x + 1} - \sqrt{lim_{x \to 0} 2 x + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + 1} - \sqrt{lim_{x \to 0} 2 x + lim_{x \to 0} 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.7

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{lim_{x \to 0} x + 1} - \sqrt{2 lim_{x \to 0} x + lim_{x \to 0} 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.8

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} x + 1} - \sqrt{2 lim_{x \to 0} x + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.9

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.1.2.9.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sqrt{0 + 1} - \sqrt{2 lim_{x \to 0} x + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.9.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sqrt{0 + 1} - \sqrt{2 \cdot 0 + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.10

Simplify the answer.

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Step 1.1.2.10.1

Simplify each term.

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Step 1.1.2.10.1.1

Add $0$ and $1$ .

$\frac{\sqrt{1} - \sqrt{2 \cdot 0 + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.10.1.2

Any root of $1$ is $1$ .

$\frac{1 - \sqrt{2 \cdot 0 + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.10.1.3

Multiply $2$ by $0$ .

$\frac{1 - \sqrt{0 + 1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.10.1.4

Add $0$ and $1$ .

$\frac{1 - \sqrt{1}}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.10.1.5

Any root of $1$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.10.1.6

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.2.10.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0} \sqrt{3 x + 4} - \sqrt{2 x + 4}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{lim_{x \to 0} \sqrt{3 x + 4} - lim_{x \to 0} \sqrt{2 x + 4}}$

Step 1.1.3.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{x \to 0} 3 x + 4} - lim_{x \to 0} \sqrt{2 x + 4}}$

Step 1.1.3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{\sqrt{lim_{x \to 0} 3 x + lim_{x \to 0} 4} - lim_{x \to 0} \sqrt{2 x + 4}}$

Step 1.1.3.4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\sqrt{3 lim_{x \to 0} x + lim_{x \to 0} 4} - lim_{x \to 0} \sqrt{2 x + 4}}$

Step 1.1.3.5

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{0}{\sqrt{3 lim_{x \to 0} x + 4} - lim_{x \to 0} \sqrt{2 x + 4}}$

Step 1.1.3.6

Move the limit under the radical sign.

$\frac{0}{\sqrt{3 lim_{x \to 0} x + 4} - \sqrt{lim_{x \to 0} 2 x + 4}}$

Step 1.1.3.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{\sqrt{3 lim_{x \to 0} x + 4} - \sqrt{lim_{x \to 0} 2 x + lim_{x \to 0} 4}}$

Step 1.1.3.8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\sqrt{3 lim_{x \to 0} x + 4} - \sqrt{2 lim_{x \to 0} x + lim_{x \to 0} 4}}$

Step 1.1.3.9

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{0}{\sqrt{3 lim_{x \to 0} x + 4} - \sqrt{2 lim_{x \to 0} x + 4}}$

Step 1.1.3.10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.1.3.10.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\sqrt{3 \cdot 0 + 4} - \sqrt{2 lim_{x \to 0} x + 4}}$

Step 1.1.3.10.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\sqrt{3 \cdot 0 + 4} - \sqrt{2 \cdot 0 + 4}}$

Step 1.1.3.11

Simplify the answer.

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Step 1.1.3.11.1

Simplify each term.

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Step 1.1.3.11.1.1

Multiply $3$ by $0$ .

$\frac{0}{\sqrt{0 + 4} - \sqrt{2 \cdot 0 + 4}}$

Step 1.1.3.11.1.2

Add $0$ and $4$ .

$\frac{0}{\sqrt{4} - \sqrt{2 \cdot 0 + 4}}$

Step 1.1.3.11.1.3

Rewrite $4$ as $2^{2}$ .

$\frac{0}{\sqrt{2^{2}} - \sqrt{2 \cdot 0 + 4}}$

Step 1.1.3.11.1.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0}{2 - \sqrt{2 \cdot 0 + 4}}$

Step 1.1.3.11.1.5

Multiply $2$ by $0$ .

$\frac{0}{2 - \sqrt{0 + 4}}$

Step 1.1.3.11.1.6

Add $0$ and $4$ .

$\frac{0}{2 - \sqrt{4}}$

Step 1.1.3.11.1.7

Rewrite $4$ as $2^{2}$ .

$\frac{0}{2 - \sqrt{2^{2}}}$

Step 1.1.3.11.1.8

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0}{2 - 1 \cdot 2}$

Step 1.1.3.11.1.9

Multiply $- 1$ by $2$ .

$\frac{0}{2 - 2}$

Step 1.1.3.11.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

Step 1.1.3.11.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.12

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{\sqrt{x + 1} - \sqrt{2 x + 1}}{\sqrt{3 x + 4} - \sqrt{2 x + 4}} = lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x + 1} - \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x + 1} - \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt{x + 1} - \sqrt{2 x + 1}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x + 1}] + \frac{d}{d x} [- \sqrt{2 x + 1}]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [\sqrt{x + 1}] + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [\sqrt{x + 1}]$ .

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Step 1.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 1}$ as ${(x + 1)}^{\frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [{(x + 1)}^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x + 1$ .

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Step 1.3.3.2.1

To apply the Chain Rule, set $u_{1}$ as $x + 1$ .

$lim_{x \to 0} \frac{\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [x + 1] + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 1] + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.2.3

Replace all occurrences of $u_{1}$ with $x + 1$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 1] + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.3

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [1]) + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} [1]) + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{\frac{1}{2} - 1} (1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.7

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.8

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.9

Simplify the numerator.

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Step 1.3.3.9.1

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{\frac{1 - 2}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.9.2

Subtract $2$ from $1$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{\frac{- 1}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.10

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{- \frac{1}{2}} (1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.11

Add $1$ and $0$ .

$lim_{x \to 0} \frac{\frac{1}{2} {(x + 1)}^{- \frac{1}{2}} \cdot 1 + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.12

Combine $\frac{1}{2}$ and ${(x + 1)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{{(x + 1)}^{- \frac{1}{2}}}{2} \cdot 1 + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.13

Multiply $\frac{{(x + 1)}^{- \frac{1}{2}}}{2}$ by $1$ .

$lim_{x \to 0} \frac{\frac{{(x + 1)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.3.14

Move ${(x + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} + \frac{d}{d x} [- \sqrt{2 x + 1}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \sqrt{2 x + 1}]$ .

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Step 1.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 1}$ as ${(2 x + 1)}^{\frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} + \frac{d}{d x} [- {(2 x + 1)}^{\frac{1}{2}}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(2 x + 1)}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(2 x + 1)}^{\frac{1}{2}}]$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{d}{d x} [{(2 x + 1)}^{\frac{1}{2}}]}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 2 x + 1$ .

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Step 1.3.4.3.1

To apply the Chain Rule, set $u_{2}$ as $2 x + 1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [2 x + 1])}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 1])}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.3.3

Replace all occurrences of $u_{2}$ with $2 x + 1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 1])}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.4

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{1}{2} - 1} (\frac{d}{d x} [2 x] + \frac{d}{d x} [1]))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{1}{2} - 1} (2 \frac{d}{d x} [x] + \frac{d}{d x} [1]))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{1}{2} - 1} (2 \cdot 1 + \frac{d}{d x} [1]))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{1}{2} - 1} (2 \cdot 1 + 0))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.9

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.10

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{1 - 1 \cdot 2}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.11

Simplify the numerator.

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Step 1.3.4.11.1

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{1 - 2}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.11.2

Subtract $2$ from $1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{\frac{- 1}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.12

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{- \frac{1}{2}} (2 \cdot 1 + 0))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.13

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{- \frac{1}{2}} (2 + 0))}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.14

Add $2$ and $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 1)}^{- \frac{1}{2}} \cdot 2)}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.15

Combine $\frac{1}{2}$ and ${(2 x + 1)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - (\frac{{(2 x + 1)}^{- \frac{1}{2}}}{2} \cdot 2)}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.16

Combine $\frac{{(2 x + 1)}^{- \frac{1}{2}}}{2}$ and $2$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{{(2 x + 1)}^{- \frac{1}{2}} \cdot 2}{2}}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.17

Move $2$ to the left of ${(2 x + 1)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{2 \cdot {(2 x + 1)}^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.18

Move ${(2 x + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{2}{2 {(2 x + 1)}^{\frac{1}{2}}}}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.19

Cancel the common factor.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{2}{2 {(2 x + 1)}^{\frac{1}{2}}}}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.4.20

Rewrite the expression.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{d}{d x} [\sqrt{3 x + 4} - \sqrt{2 x + 4}]}$

Step 1.3.5

By the Sum Rule, the derivative of $\sqrt{3 x + 4} - \sqrt{2 x + 4}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{3 x + 4}] + \frac{d}{d x} [- \sqrt{2 x + 4}]$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{d}{d x} [\sqrt{3 x + 4}] + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6

Evaluate $\frac{d}{d x} [\sqrt{3 x + 4}]$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{d}{d x} [{(3 x + 4)}^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 3 x + 4$ .

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Step 1.3.6.2.1

To apply the Chain Rule, set $u_{1}$ as $3 x + 4$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [3 x + 4] + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [3 x + 4] + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.2.3

Replace all occurrences of $u_{1}$ with $3 x + 4$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{1}{2} - 1} \frac{d}{d x} [3 x + 4] + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.3

By the Sum Rule, the derivative of $3 x + 4$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [4]$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{1}{2} - 1} (\frac{d}{d x} [3 x] + \frac{d}{d x} [4]) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{1}{2} - 1} (3 \frac{d}{d x} [x] + \frac{d}{d x} [4]) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{1}{2} - 1} (3 \cdot 1 + \frac{d}{d x} [4]) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{1}{2} - 1} (3 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (3 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.8

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (3 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.9

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{1 - 1 \cdot 2}{2}} (3 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.10

Simplify the numerator.

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Step 1.3.6.10.1

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{1 - 2}{2}} (3 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.10.2

Subtract $2$ from $1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{\frac{- 1}{2}} (3 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.11

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{- \frac{1}{2}} (3 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.12

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{- \frac{1}{2}} (3 + 0) + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.13

Add $3$ and $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{1}{2} {(3 x + 4)}^{- \frac{1}{2}} \cdot 3 + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.14

Combine $\frac{1}{2}$ and ${(3 x + 4)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{{(3 x + 4)}^{- \frac{1}{2}}}{2} \cdot 3 + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.15

Combine $\frac{{(3 x + 4)}^{- \frac{1}{2}}}{2}$ and $3$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{{(3 x + 4)}^{- \frac{1}{2}} \cdot 3}{2} + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.16

Move $3$ to the left of ${(3 x + 4)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3 \cdot {(3 x + 4)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.6.17

Move ${(3 x + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} + \frac{d}{d x} [- \sqrt{2 x + 4}]}$

Step 1.3.7

Evaluate $\frac{d}{d x} [- \sqrt{2 x + 4}]$ .

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Step 1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 4}$ as ${(2 x + 4)}^{\frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} + \frac{d}{d x} [- {(2 x + 4)}^{\frac{1}{2}}]}$

Step 1.3.7.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(2 x + 4)}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(2 x + 4)}^{\frac{1}{2}}]$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - \frac{d}{d x} [{(2 x + 4)}^{\frac{1}{2}}]}$

Step 1.3.7.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 2 x + 4$ .

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Step 1.3.7.3.1

To apply the Chain Rule, set $u_{2}$ as $2 x + 4$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [2 x + 4])}$

Step 1.3.7.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 4])}$

Step 1.3.7.3.3

Replace all occurrences of $u_{2}$ with $2 x + 4$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 4)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 4])}$

Step 1.3.7.4

By the Sum Rule, the derivative of $2 x + 4$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [4]$ .

Step 1.3.7.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

Step 1.3.7.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

Step 1.3.7.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 4)}^{\frac{1}{2} - 1} (2 \cdot 1 + 0))}$

Step 1.3.7.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

Step 1.3.7.9

Combine $- 1$ and $\frac{2}{2}$ .

Step 1.3.7.10

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 4)}^{\frac{1 - 1 \cdot 2}{2}} (2 \cdot 1 + 0))}$

Step 1.3.7.11

Simplify the numerator.

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Step 1.3.7.11.1

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 4)}^{\frac{1 - 2}{2}} (2 \cdot 1 + 0))}$

Step 1.3.7.11.2

Subtract $2$ from $1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 4)}^{\frac{- 1}{2}} (2 \cdot 1 + 0))}$

Step 1.3.7.12

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 4)}^{- \frac{1}{2}} (2 \cdot 1 + 0))}$

Step 1.3.7.13

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 4)}^{- \frac{1}{2}} (2 + 0))}$

Step 1.3.7.14

Add $2$ and $0$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{1}{2} {(2 x + 4)}^{- \frac{1}{2}} \cdot 2)}$

Step 1.3.7.15

Combine $\frac{1}{2}$ and ${(2 x + 4)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - (\frac{{(2 x + 4)}^{- \frac{1}{2}}}{2} \cdot 2)}$

Step 1.3.7.16

Combine $\frac{{(2 x + 4)}^{- \frac{1}{2}}}{2}$ and $2$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - \frac{{(2 x + 4)}^{- \frac{1}{2}} \cdot 2}{2}}$

Step 1.3.7.17

Move $2$ to the left of ${(2 x + 4)}^{- \frac{1}{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - \frac{2 \cdot {(2 x + 4)}^{- \frac{1}{2}}}{2}}$

Step 1.3.7.18

Move ${(2 x + 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - \frac{2}{2 {(2 x + 4)}^{\frac{1}{2}}}}$

Step 1.3.7.19

Cancel the common factor.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - \frac{2}{2 {(2 x + 4)}^{\frac{1}{2}}}}$

Step 1.3.7.20

Rewrite the expression.

$lim_{x \to 0} \frac{\frac{1}{2 {(x + 1)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 4)}^{\frac{1}{2}}}}$

Step 1.4

Convert fractional exponents to radicals.

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Step 1.4.1

Rewrite ${(x + 1)}^{\frac{1}{2}}$ as $\sqrt{x + 1}$ .

$lim_{x \to 0} \frac{\frac{1}{2 \sqrt{x + 1}} - \frac{1}{{(2 x + 1)}^{\frac{1}{2}}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 4)}^{\frac{1}{2}}}}$

Step 1.4.2

Rewrite ${(2 x + 1)}^{\frac{1}{2}}$ as $\sqrt{2 x + 1}$ .

$lim_{x \to 0} \frac{\frac{1}{2 \sqrt{x + 1}} - \frac{1}{\sqrt{2 x + 1}}}{\frac{3}{2 {(3 x + 4)}^{\frac{1}{2}}} - \frac{1}{{(2 x + 4)}^{\frac{1}{2}}}}$

Step 1.4.3

Rewrite ${(3 x + 4)}^{\frac{1}{2}}$ as $\sqrt{3 x + 4}$ .

$lim_{x \to 0} \frac{\frac{1}{2 \sqrt{x + 1}} - \frac{1}{\sqrt{2 x + 1}}}{\frac{3}{2 \sqrt{3 x + 4}} - \frac{1}{{(2 x + 4)}^{\frac{1}{2}}}}$

Step 1.4.4

Rewrite ${(2 x + 4)}^{\frac{1}{2}}$ as $\sqrt{2 x + 4}$ .

$lim_{x \to 0} \frac{\frac{1}{2 \sqrt{x + 1}} - \frac{1}{\sqrt{2 x + 1}}}{\frac{3}{2 \sqrt{3 x + 4}} - \frac{1}{\sqrt{2 x + 4}}}$

Step 1.5

Combine terms.

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Step 1.5.1

To write $\frac{1}{2 \sqrt{x + 1}}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{2 x + 1}}{\sqrt{2 x + 1}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 \sqrt{x + 1}} \cdot \frac{\sqrt{2 x + 1}}{\sqrt{2 x + 1}} - \frac{1}{\sqrt{2 x + 1}}}{\frac{3}{2 \sqrt{3 x + 4}} - \frac{1}{\sqrt{2 x + 4}}}$

Step 1.5.2

To write $- \frac{1}{\sqrt{2 x + 1}}$ as a fraction with a common denominator, multiply by $\frac{2 \sqrt{x + 1}}{2 \sqrt{x + 1}}$ .

$lim_{x \to 0} \frac{\frac{1}{2 \sqrt{x + 1}} \cdot \frac{\sqrt{2 x + 1}}{\sqrt{2 x + 1}} - \frac{1}{\sqrt{2 x + 1}} \cdot \frac{2 \sqrt{x + 1}}{2 \sqrt{x + 1}}}{\frac{3}{2 \sqrt{3 x + 4}} - \frac{1}{\sqrt{2 x + 4}}}$

Step 1.5.3

Write each expression with a common denominator of $2 \sqrt{x + 1} \sqrt{2 x + 1}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.1

Multiply $\frac{1}{2 \sqrt{x + 1}}$ by $\frac{\sqrt{2 x + 1}}{\sqrt{2 x + 1}}$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1}}{2 \sqrt{x + 1} \sqrt{2 x + 1}} - \frac{1}{\sqrt{2 x + 1}} \cdot \frac{2 \sqrt{x + 1}}{2 \sqrt{x + 1}}}{\frac{3}{2 \sqrt{3 x + 4}} - \frac{1}{\sqrt{2 x + 4}}}$

Step 1.5.3.2

Combine using the product rule for radicals.

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}} - \frac{1}{\sqrt{2 x + 1}} \cdot \frac{2 \sqrt{x + 1}}{2 \sqrt{x + 1}}}{\frac{3}{2 \sqrt{3 x + 4}} - \frac{1}{\sqrt{2 x + 4}}}$

Step 1.5.3.3

Multiply $\frac{1}{\sqrt{2 x + 1}}$ by $\frac{2 \sqrt{x + 1}}{2 \sqrt{x + 1}}$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}} - \frac{2 \sqrt{x + 1}}{\sqrt{2 x + 1} (2 \sqrt{x + 1})}}{\frac{3}{2 \sqrt{3 x + 4}} - \frac{1}{\sqrt{2 x + 4}}}$

Step 1.5.3.4

Combine using the product rule for radicals.

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}} - \frac{2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3}{2 \sqrt{3 x + 4}} - \frac{1}{\sqrt{2 x + 4}}}$

Step 1.5.4

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3}{2 \sqrt{3 x + 4}} - \frac{1}{\sqrt{2 x + 4}}}$

Step 1.5.5

To write $\frac{3}{2 \sqrt{3 x + 4}}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{2 x + 4}}{\sqrt{2 x + 4}}$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3}{2 \sqrt{3 x + 4}} \cdot \frac{\sqrt{2 x + 4}}{\sqrt{2 x + 4}} - \frac{1}{\sqrt{2 x + 4}}}$

Step 1.5.6

To write $- \frac{1}{\sqrt{2 x + 4}}$ as a fraction with a common denominator, multiply by $\frac{2 \sqrt{3 x + 4}}{2 \sqrt{3 x + 4}}$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3}{2 \sqrt{3 x + 4}} \cdot \frac{\sqrt{2 x + 4}}{\sqrt{2 x + 4}} - \frac{1}{\sqrt{2 x + 4}} \cdot \frac{2 \sqrt{3 x + 4}}{2 \sqrt{3 x + 4}}}$

Step 1.5.7

Write each expression with a common denominator of $2 \sqrt{3 x + 4} \sqrt{2 x + 4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.7.1

Multiply $\frac{3}{2 \sqrt{3 x + 4}}$ by $\frac{\sqrt{2 x + 4}}{\sqrt{2 x + 4}}$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{3 x + 4} \sqrt{2 x + 4}} - \frac{1}{\sqrt{2 x + 4}} \cdot \frac{2 \sqrt{3 x + 4}}{2 \sqrt{3 x + 4}}}$

Step 1.5.7.2

Combine using the product rule for radicals.

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{(2 x + 4) (3 x + 4)}} - \frac{1}{\sqrt{2 x + 4}} \cdot \frac{2 \sqrt{3 x + 4}}{2 \sqrt{3 x + 4}}}$

Step 1.5.7.3

Factor $2$ out of $2 x + 4$ .

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Step 1.5.7.3.1

Factor $2$ out of $2 x$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{(2 (x) + 4) (3 x + 4)}} - \frac{1}{\sqrt{2 x + 4}} \cdot \frac{2 \sqrt{3 x + 4}}{2 \sqrt{3 x + 4}}}$

Step 1.5.7.3.2

Factor $2$ out of $4$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{(2 x + 2 \cdot 2) (3 x + 4)}} - \frac{1}{\sqrt{2 x + 4}} \cdot \frac{2 \sqrt{3 x + 4}}{2 \sqrt{3 x + 4}}}$

Step 1.5.7.3.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}} - \frac{1}{\sqrt{2 x + 4}} \cdot \frac{2 \sqrt{3 x + 4}}{2 \sqrt{3 x + 4}}}$

Step 1.5.7.4

Multiply $\frac{1}{\sqrt{2 x + 4}}$ by $\frac{2 \sqrt{3 x + 4}}{2 \sqrt{3 x + 4}}$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}} - \frac{2 \sqrt{3 x + 4}}{\sqrt{2 x + 4} (2 \sqrt{3 x + 4})}}$

Step 1.5.7.5

Combine using the product rule for radicals.

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}} - \frac{2 \sqrt{3 x + 4}}{2 \sqrt{(2 x + 4) (3 x + 4)}}}$

Step 1.5.7.6

Factor $2$ out of $2 x + 4$ .

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Step 1.5.7.6.1

Factor $2$ out of $2 x$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}} - \frac{2 \sqrt{3 x + 4}}{2 \sqrt{(2 (x) + 4) (3 x + 4)}}}$

Step 1.5.7.6.2

Factor $2$ out of $4$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}} - \frac{2 \sqrt{3 x + 4}}{2 \sqrt{(2 x + 2 \cdot 2) (3 x + 4)}}}$

Step 1.5.7.6.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}} - \frac{2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 1.5.8

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{\frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{2 \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} lim_{x \to 0} \frac{\sqrt{2 x + 1} - 2 \sqrt{x + 1}}{\sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{lim_{x \to 0} \sqrt{2 x + 1} - 2 \sqrt{x + 1}}{lim_{x \to 0} \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{lim_{x \to 0} \sqrt{2 x + 1} - lim_{x \to 0} 2 \sqrt{x + 1}}{lim_{x \to 0} \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.5

Move the limit under the radical sign.

$\frac{\frac{1}{2} \cdot \frac{\sqrt{lim_{x \to 0} 2 x + 1} - lim_{x \to 0} 2 \sqrt{x + 1}}{lim_{x \to 0} \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{lim_{x \to 0} 2 x + lim_{x \to 0} 1} - lim_{x \to 0} 2 \sqrt{x + 1}}{lim_{x \to 0} \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.7

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + lim_{x \to 0} 1} - lim_{x \to 0} 2 \sqrt{x + 1}}{lim_{x \to 0} \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.8

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - lim_{x \to 0} 2 \sqrt{x + 1}}{lim_{x \to 0} \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.9

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 lim_{x \to 0} \sqrt{x + 1}}{lim_{x \to 0} \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.10

Move the limit under the radical sign.

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{lim_{x \to 0} \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.11

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + lim_{x \to 0} 1}}{lim_{x \to 0} \sqrt{(2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.12

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

Step 2.13

Move the limit under the radical sign.

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{lim_{x \to 0} (2 x + 1) (x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.14

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{lim_{x \to 0} 2 x + 1 \cdot lim_{x \to 0} x + 1}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.15

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(lim_{x \to 0} 2 x + lim_{x \to 0} 1) \cdot lim_{x \to 0} x + 1}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.16

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + lim_{x \to 0} 1) \cdot lim_{x \to 0} x + 1}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.17

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot lim_{x \to 0} x + 1}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.18

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + lim_{x \to 0} 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.19

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{2 \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.20

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} lim_{x \to 0} \frac{3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{\sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.21

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{lim_{x \to 0} 3 \sqrt{2 x + 4} - 2 \sqrt{3 x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.22

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{lim_{x \to 0} 3 \sqrt{2 x + 4} - lim_{x \to 0} 2 \sqrt{3 x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.23

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 lim_{x \to 0} \sqrt{2 x + 4} - lim_{x \to 0} 2 \sqrt{3 x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.24

Move the limit under the radical sign.

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{lim_{x \to 0} 2 x + 4} - lim_{x \to 0} 2 \sqrt{3 x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.25

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{lim_{x \to 0} 2 x + lim_{x \to 0} 4} - lim_{x \to 0} 2 \sqrt{3 x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.26

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + lim_{x \to 0} 4} - lim_{x \to 0} 2 \sqrt{3 x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.27

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - lim_{x \to 0} 2 \sqrt{3 x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.28

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 lim_{x \to 0} \sqrt{3 x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.29

Move the limit under the radical sign.

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{lim_{x \to 0} 3 x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.30

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{lim_{x \to 0} 3 x + lim_{x \to 0} 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.31

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + lim_{x \to 0} 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.32

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{lim_{x \to 0} \sqrt{2 (x + 2) (3 x + 4)}}}$

Step 2.33

Move the limit under the radical sign.

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{lim_{x \to 0} 2 (x + 2) (3 x + 4)}}}$

Step 2.34

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 lim_{x \to 0} (x + 2) (3 x + 4)}}}$

Step 2.35

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 lim_{x \to 0} x + 2 \cdot lim_{x \to 0} 3 x + 4}}}$

Step 2.36

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + lim_{x \to 0} 2) \cdot lim_{x \to 0} 3 x + 4}}}$

Step 2.37

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot lim_{x \to 0} 3 x + 4}}}$

Step 2.38

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot (lim_{x \to 0} 3 x + lim_{x \to 0} 4)}}}$

Step 2.39

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot (3 lim_{x \to 0} x + lim_{x \to 0} 4)}}}$

Step 2.40

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 lim_{x \to 0} x + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot (3 lim_{x \to 0} x + 4)}}}$

Step 3

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 \cdot 0 + 1} - 2 \sqrt{lim_{x \to 0} x + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot (3 lim_{x \to 0} x + 4)}}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 \cdot 0 + 1} - 2 \sqrt{0 + 1}}{\sqrt{(2 lim_{x \to 0} x + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot (3 lim_{x \to 0} x + 4)}}}$

Step 3.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 \cdot 0 + 1} - 2 \sqrt{0 + 1}}{\sqrt{(2 \cdot 0 + 1) \cdot (lim_{x \to 0} x + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot (3 lim_{x \to 0} x + 4)}}}$

Step 3.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 \cdot 0 + 1} - 2 \sqrt{0 + 1}}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 lim_{x \to 0} x + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot (3 lim_{x \to 0} x + 4)}}}$

Step 3.5

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 \cdot 0 + 1} - 2 \sqrt{0 + 1}}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 lim_{x \to 0} x + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot (3 lim_{x \to 0} x + 4)}}}$

Step 3.6

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2 \cdot 0 + 1} - 2 \sqrt{0 + 1}}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}}}{\frac{1}{2} \cdot \frac{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}{\sqrt{2 (lim_{x \to 0} x + 2) \cdot (3 lim_{x \to 0} x + 4)}}}$

Step 3.7

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

Step 3.8

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

Step 4

Simplify the answer.

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Step 4.1

Cancel the common factor of $\frac{1}{2}$ .

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Step 4.1.1

Cancel the common factor.

Step 4.1.2

Rewrite the expression.

$\frac{\frac{\sqrt{2 \cdot 0 + 1} - 2 \sqrt{0 + 1}}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}}}{\frac{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{2 \cdot 0 + 1} - 2 \sqrt{0 + 1}}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Multiply $2$ by $0$ .

$\frac{\sqrt{0 + 1} - 2 \sqrt{0 + 1}}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.3.2

Add $0$ and $1$ .

$\frac{\sqrt{1} - 2 \sqrt{0 + 1}}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.3.3

Any root of $1$ is $1$ .

$\frac{1 - 2 \sqrt{0 + 1}}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.3.4

Add $0$ and $1$ .

$\frac{1 - 2 \sqrt{1}}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.3.5

Any root of $1$ is $1$ .

$\frac{1 - 2 \cdot 1}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.3.6

Multiply $- 2$ by $1$ .

$\frac{1 - 2}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.3.7

Subtract $2$ from $1$ .

$\frac{- 1}{\sqrt{(2 \cdot 0 + 1) \cdot (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.4

Simplify the denominator.

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Step 4.4.1

Multiply $2$ by $0$ .

$\frac{- 1}{\sqrt{(0 + 1) (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.4.2

Add $0$ and $1$ .

$\frac{- 1}{\sqrt{1 (0 + 1)}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.4.3

Multiply $0 + 1$ by $1$ .

$\frac{- 1}{\sqrt{0 + 1}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.4.4

Add $0$ and $1$ .

$\frac{- 1}{\sqrt{1}} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.4.5

Any root of $1$ is $1$ .

$\frac{- 1}{1} \cdot \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.5

Divide $- 1$ by $1$ .

$- \frac{\sqrt{2 (0 + 2) \cdot (3 \cdot 0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $3$ by $0$ .

$- \frac{\sqrt{2 (0 + 2) (0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.6.2

Add $0$ and $2$ .

$- \frac{\sqrt{2 \cdot 2 (0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.6.3

Multiply $2$ by $2$ .

$- \frac{\sqrt{4 (0 + 4)}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.6.4

Add $0$ and $4$ .

$- \frac{\sqrt{4 \cdot 4}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.6.5

Multiply $4$ by $4$ .

$- \frac{\sqrt{16}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.6.6

Rewrite $16$ as $4^{2}$ .

$- \frac{\sqrt{4^{2}}}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.6.7

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{4}{3 \sqrt{2 \cdot 0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.7

Simplify the denominator.

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Step 4.7.1

Multiply $2$ by $0$ .

$- \frac{4}{3 \sqrt{0 + 4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.7.2

Add $0$ and $4$ .

$- \frac{4}{3 \sqrt{4} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.7.3

Rewrite $4$ as $2^{2}$ .

$- \frac{4}{3 \sqrt{2^{2}} - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.7.4

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{4}{3 \cdot 2 - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.7.5

Multiply $3$ by $2$ .

$- \frac{4}{6 - 2 \sqrt{3 \cdot 0 + 4}}$

Step 4.7.6

Multiply $3$ by $0$ .

$- \frac{4}{6 - 2 \sqrt{0 + 4}}$

Step 4.7.7

Add $0$ and $4$ .

$- \frac{4}{6 - 2 \sqrt{4}}$

Step 4.7.8

Rewrite $4$ as $2^{2}$ .

$- \frac{4}{6 - 2 \sqrt{2^{2}}}$

Step 4.7.9

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{4}{6 - 2 \cdot 2}$

Step 4.7.10

Multiply $- 2$ by $2$ .

$- \frac{4}{6 - 4}$

Step 4.7.11

Subtract $4$ from $6$ .

$- \frac{4}{2}$

Step 4.8

Divide $4$ by $2$ .

$- 1 \cdot 2$

Step 4.9

Multiply $- 1$ by $2$ .

$- 2$