Calculus Examples

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

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.

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

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 $0$ .

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

Step 1.1.2.2

Move the limit under the radical sign.

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

Move the limit under the radical sign.

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

Step 1.1.2.7

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

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

Step 1.1.2.8

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

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

Step 1.1.2.9

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.1.2.10

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

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Step 1.1.2.10.1

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

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

Step 1.1.2.10.2

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

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

Step 1.1.2.11

Simplify the answer.

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Step 1.1.2.11.1

Simplify each term.

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Step 1.1.2.11.1.1

Multiply $2$ by $0$ .

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

Step 1.1.2.11.1.2

Add $1$ and $0$ .

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

Step 1.1.2.11.1.3

Any root of $1$ is $1$ .

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

Step 1.1.2.11.1.4

Multiply $- 4$ by $0$ .

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

Step 1.1.2.11.1.5

Add $1$ and $0$ .

$\frac{1 - \sqrt{1}}{lim_{x \to 0} x}$

Step 1.1.2.11.1.6

Any root of $1$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to 0} x}$

Step 1.1.2.11.1.7

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 0} x}$

Step 1.1.2.11.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0} x}$

Step 1.1.3

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

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt{1 + 2 x} - \sqrt{1 - 4 x}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{1 + 2 x}] + \frac{d}{d x} [- \sqrt{1 - 4 x}]$ .

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

Step 1.3.3

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

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Step 1.3.3.1

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

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

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 $g (x) = 1 + 2 x$ .

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Step 1.3.3.2.1

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

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

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 $n = \frac{1}{2}$ .

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

Step 1.3.3.2.3

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

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

Step 1.3.3.3

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

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

Step 1.3.3.4

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

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

Step 1.3.3.5

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

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

Step 1.3.3.6

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

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

Step 1.3.3.7

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

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

Step 1.3.3.8

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

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

Step 1.3.3.9

Combine the numerators over the common denominator.

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

Step 1.3.3.10

Simplify the numerator.

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Step 1.3.3.10.1

Multiply $- 1$ by $2$ .

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

Step 1.3.3.10.2

Subtract $2$ from $1$ .

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

Step 1.3.3.11

Move the negative in front of the fraction.

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

Step 1.3.3.12

Multiply $2$ by $1$ .

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

Step 1.3.3.13

Add $0$ and $2$ .

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

Step 1.3.3.14

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

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

Step 1.3.3.15

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

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

Step 1.3.3.16

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

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

Step 1.3.3.17

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

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

Step 1.3.3.18

Cancel the common factor.

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

Step 1.3.3.19

Rewrite the expression.

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

Step 1.3.4

Evaluate $\frac{d}{d x} [- \sqrt{1 - 4 x}]$ .

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

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 $g (x) = 1 - 4 x$ .

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Step 1.3.4.3.1

To apply the Chain Rule, set $u_{2}$ as $1 - 4 x$ .

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

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 $n = \frac{1}{2}$ .

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

Step 1.3.4.3.3

Replace all occurrences of $u_{2}$ with $1 - 4 x$ .

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

Step 1.3.4.4

By the Sum Rule, the derivative of $1 - 4 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x]$ .

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

Step 1.3.4.5

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

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

Step 1.3.4.6

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

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

Step 1.3.4.7

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

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

Step 1.3.4.8

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

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

Step 1.3.4.9

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

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

Step 1.3.4.10

Combine the numerators over the common denominator.

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

Step 1.3.4.11

Simplify the numerator.

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Step 1.3.4.11.1

Multiply $- 1$ by $2$ .

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

Step 1.3.4.11.2

Subtract $2$ from $1$ .

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

Step 1.3.4.12

Move the negative in front of the fraction.

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

Step 1.3.4.13

Multiply $- 4$ by $1$ .

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

Step 1.3.4.14

Subtract $4$ from $0$ .

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

Step 1.3.4.15

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

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

Step 1.3.4.16

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

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

Step 1.3.4.17

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

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

Step 1.3.4.18

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

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

Step 1.3.4.19

Factor $2$ out of $- 4$ .

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

Step 1.3.4.20

Cancel the common factors.

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Step 1.3.4.20.1

Factor $2$ out of $2 {(1 - 4 x)}^{\frac{1}{2}}$ .

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

Step 1.3.4.20.2

Cancel the common factor.

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

Step 1.3.4.20.3

Rewrite the expression.

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

Step 1.3.4.21

Move the negative in front of the fraction.

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

Step 1.3.4.22

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4.23

Multiply $\frac{2}{{(1 - 4 x)}^{\frac{1}{2}}}$ by $1$ .

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

Step 1.3.5

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

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

Step 1.4

Convert fractional exponents to radicals.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.5

Divide $\frac{1}{\sqrt{1 + 2 x}} + \frac{2}{\sqrt{1 - 4 x}}$ by $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Move the limit under the radical sign.

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Move the limit under the radical sign.

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

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 $x$ .

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

Step 3.2

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Simplify the denominator.

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Step 4.1.1.1

Multiply $2$ by $0$ .

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

Step 4.1.1.2

Add $1$ and $0$ .

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

Step 4.1.1.3

Any root of $1$ is $1$ .

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

Step 4.1.2

Divide $1$ by $1$ .

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

Step 4.1.3

Simplify the denominator.

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Step 4.1.3.1

Multiply $- 4$ by $0$ .

$1 + 2 \frac{1}{\sqrt{1 + 0}}$

Step 4.1.3.2

Add $1$ and $0$ .

$1 + 2 \frac{1}{\sqrt{1}}$

Step 4.1.3.3

Any root of $1$ is $1$ .

$1 + 2 (\frac{1}{1})$

Step 4.1.4

Cancel the common factor of $1$ .

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Step 4.1.4.1

Cancel the common factor.

$1 + 2 (\frac{1}{1})$

Step 4.1.4.2

Rewrite the expression.

$1 + 2 \cdot 1$

Step 4.1.5

Multiply $2$ by $1$ .

$1 + 2$

Step 4.2

Add $1$ and $2$ .

$3$