Calculus Examples

$lim_{x \to 2} \frac{\sqrt{2 x + 5} - \sqrt{x + 7}}{x - 2}$

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.

$\frac{lim_{x \to 2} \sqrt{2 x + 5} - \sqrt{x + 7}}{lim_{x \to 2} x - 2}$

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 $2$ .

$\frac{lim_{x \to 2} \sqrt{2 x + 5} - lim_{x \to 2} \sqrt{x + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 2} 2 x + 5} - lim_{x \to 2} \sqrt{x + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{\sqrt{lim_{x \to 2} 2 x + lim_{x \to 2} 5} - lim_{x \to 2} \sqrt{x + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.4

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

$\frac{\sqrt{2 lim_{x \to 2} x + lim_{x \to 2} 5} - lim_{x \to 2} \sqrt{x + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.5

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{2 lim_{x \to 2} x + 5} - lim_{x \to 2} \sqrt{x + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.6

Move the limit under the radical sign.

$\frac{\sqrt{2 lim_{x \to 2} x + 5} - \sqrt{lim_{x \to 2} x + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{\sqrt{2 lim_{x \to 2} x + 5} - \sqrt{lim_{x \to 2} x + lim_{x \to 2} 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.8

Evaluate the limit of $7$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{2 lim_{x \to 2} x + 5} - \sqrt{lim_{x \to 2} x + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.9

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 1.1.2.9.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2 \cdot 2 + 5} - \sqrt{lim_{x \to 2} x + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.9.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2 \cdot 2 + 5} - \sqrt{2 + 7}}{lim_{x \to 2} x - 2}$

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

Multiply $2$ by $2$ .

$\frac{\sqrt{4 + 5} - \sqrt{2 + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.10.1.2

Add $4$ and $5$ .

$\frac{\sqrt{9} - \sqrt{2 + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.10.1.3

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}} - \sqrt{2 + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.10.1.4

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

$\frac{3 - \sqrt{2 + 7}}{lim_{x \to 2} x - 2}$

Step 1.1.2.10.1.5

Add $2$ and $7$ .

$\frac{3 - \sqrt{9}}{lim_{x \to 2} x - 2}$

Step 1.1.2.10.1.6

Rewrite $9$ as $3^{2}$ .

$\frac{3 - \sqrt{3^{2}}}{lim_{x \to 2} x - 2}$

Step 1.1.2.10.1.7

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

$\frac{3 - 1 \cdot 3}{lim_{x \to 2} x - 2}$

Step 1.1.2.10.1.8

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{lim_{x \to 2} x - 2}$

Step 1.1.2.10.2

Subtract $3$ from $3$ .

$\frac{0}{lim_{x \to 2} x - 2}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{0}{lim_{x \to 2} x - lim_{x \to 2} 2}$

Step 1.1.3.1.2

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $2$ .

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

Step 1.1.3.3.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

$lim_{x \to 2} \frac{\sqrt{2 x + 5} - \sqrt{x + 7}}{x - 2} = lim_{x \to 2} \frac{\frac{d}{d x} [\sqrt{2 x + 5} - \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 2} \frac{\frac{d}{d x} [\sqrt{2 x + 5} - \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt{2 x + 5} - \sqrt{x + 7}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{2 x + 5}] + \frac{d}{d x} [- \sqrt{x + 7}]$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [\sqrt{2 x + 5}] + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [\sqrt{2 x + 5}]$ .

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Step 1.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 5}$ as ${(2 x + 5)}^{\frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

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 $g (x) = 2 x + 5$ .

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Step 1.3.3.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x + 5$ .

$lim_{x \to 2} \frac{\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [2 x + 5] + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

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

$lim_{x \to 2} \frac{\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 5] + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.2.3

Replace all occurrences of $u_{1}$ with $2 x + 5$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 5] + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.3

By the Sum Rule, the derivative of $2 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} (\frac{d}{d x} [2 x] + \frac{d}{d x} [5]) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.4

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

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} (2 \frac{d}{d x} [x] + \frac{d}{d x} [5]) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.5

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

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} (2 \cdot 1 + \frac{d}{d x} [5]) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.6

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} (2 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.7

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

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (2 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.8

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

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (2 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.9

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{1 - 1 \cdot 2}{2}} (2 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.10

Simplify the numerator.

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Step 1.3.3.10.1

Multiply $- 1$ by $2$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{1 - 2}{2}} (2 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.10.2

Subtract $2$ from $1$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{\frac{- 1}{2}} (2 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.11

Move the negative in front of the fraction.

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{- \frac{1}{2}} (2 \cdot 1 + 0) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.12

Multiply $2$ by $1$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{- \frac{1}{2}} (2 + 0) + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.13

Add $2$ and $0$ .

$lim_{x \to 2} \frac{\frac{1}{2} {(2 x + 5)}^{- \frac{1}{2}} \cdot 2 + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.14

Combine $\frac{1}{2}$ and ${(2 x + 5)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{{(2 x + 5)}^{- \frac{1}{2}}}{2} \cdot 2 + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.15

Combine $\frac{{(2 x + 5)}^{- \frac{1}{2}}}{2}$ and $2$ .

$lim_{x \to 2} \frac{\frac{{(2 x + 5)}^{- \frac{1}{2}} \cdot 2}{2} + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.16

Move $2$ to the left of ${(2 x + 5)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{2 \cdot {(2 x + 5)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.17

Move ${(2 x + 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 2} \frac{\frac{2}{2 {(2 x + 5)}^{\frac{1}{2}}} + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.18

Cancel the common factor.

$lim_{x \to 2} \frac{\frac{2}{2 {(2 x + 5)}^{\frac{1}{2}}} + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.3.19

Rewrite the expression.

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} + \frac{d}{d x} [- \sqrt{x + 7}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \sqrt{x + 7}]$ .

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Step 1.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} + \frac{d}{d x} [- {(x + 7)}^{\frac{1}{2}}]}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(x + 7)}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(x + 7)}^{\frac{1}{2}}]$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - \frac{d}{d x} [{(x + 7)}^{\frac{1}{2}}]}{\frac{d}{d x} [x - 2]}$

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 $g (x) = x + 7$ .

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Step 1.3.4.3.1

To apply the Chain Rule, set $u_{2}$ as $x + 7$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [x + 7])}{\frac{d}{d x} [x - 2]}$

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

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 7])}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.3.3

Replace all occurrences of $u_{2}$ with $x + 7$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 7])}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.4

By the Sum Rule, the derivative of $x + 7$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [7]$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [7]))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.5

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

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} [7]))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.6

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{\frac{1}{2} - 1} (1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.7

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

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.8

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

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.9

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.10

Simplify the numerator.

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Step 1.3.4.10.1

Multiply $- 1$ by $2$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{\frac{1 - 2}{2}} (1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.10.2

Subtract $2$ from $1$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{\frac{- 1}{2}} (1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.11

Move the negative in front of the fraction.

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{- \frac{1}{2}} (1 + 0))}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.12

Add $1$ and $0$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{1}{2} {(x + 7)}^{- \frac{1}{2}} \cdot 1)}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.13

Combine $\frac{1}{2}$ and ${(x + 7)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - (\frac{{(x + 7)}^{- \frac{1}{2}}}{2} \cdot 1)}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.14

Multiply $\frac{{(x + 7)}^{- \frac{1}{2}}}{2}$ by $1$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - \frac{{(x + 7)}^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [x - 2]}$

Step 1.3.4.15

Move ${(x + 7)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - \frac{1}{2 {(x + 7)}^{\frac{1}{2}}}}{\frac{d}{d x} [x - 2]}$

Step 1.3.5

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

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - \frac{1}{2 {(x + 7)}^{\frac{1}{2}}}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 2]}$

Step 1.3.6

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

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - \frac{1}{2 {(x + 7)}^{\frac{1}{2}}}}{1 + \frac{d}{d x} [- 2]}$

Step 1.3.7

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - \frac{1}{2 {(x + 7)}^{\frac{1}{2}}}}{1 + 0}$

Step 1.3.8

Add $1$ and $0$ .

$lim_{x \to 2} \frac{\frac{1}{{(2 x + 5)}^{\frac{1}{2}}} - \frac{1}{2 {(x + 7)}^{\frac{1}{2}}}}{1}$

Step 1.4

Convert fractional exponents to radicals.

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Step 1.4.1

Rewrite ${(2 x + 5)}^{\frac{1}{2}}$ as $\sqrt{2 x + 5}$ .

$lim_{x \to 2} \frac{\frac{1}{\sqrt{2 x + 5}} - \frac{1}{2 {(x + 7)}^{\frac{1}{2}}}}{1}$

Step 1.4.2

Rewrite ${(x + 7)}^{\frac{1}{2}}$ as $\sqrt{x + 7}$ .

$lim_{x \to 2} \frac{\frac{1}{\sqrt{2 x + 5}} - \frac{1}{2 \sqrt{x + 7}}}{1}$

Step 1.5

Combine terms.

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Step 1.5.1

To write $\frac{1}{\sqrt{2 x + 5}}$ as a fraction with a common denominator, multiply by $\frac{2 \sqrt{x + 7}}{2 \sqrt{x + 7}}$ .

$lim_{x \to 2} \frac{\frac{1}{\sqrt{2 x + 5}} \cdot \frac{2 \sqrt{x + 7}}{2 \sqrt{x + 7}} - \frac{1}{2 \sqrt{x + 7}}}{1}$

Step 1.5.2

To write $- \frac{1}{2 \sqrt{x + 7}}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{2 x + 5}}{\sqrt{2 x + 5}}$ .

$lim_{x \to 2} \frac{\frac{1}{\sqrt{2 x + 5}} \cdot \frac{2 \sqrt{x + 7}}{2 \sqrt{x + 7}} - \frac{1}{2 \sqrt{x + 7}} \cdot \frac{\sqrt{2 x + 5}}{\sqrt{2 x + 5}}}{1}$

Step 1.5.3

Write each expression with a common denominator of $\sqrt{2 x + 5} \cdot 2 \sqrt{x + 7}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.1

Multiply $\frac{1}{\sqrt{2 x + 5}}$ by $\frac{2 \sqrt{x + 7}}{2 \sqrt{x + 7}}$ .

$lim_{x \to 2} \frac{\frac{2 \sqrt{x + 7}}{\sqrt{2 x + 5} (2 \sqrt{x + 7})} - \frac{1}{2 \sqrt{x + 7}} \cdot \frac{\sqrt{2 x + 5}}{\sqrt{2 x + 5}}}{1}$

Step 1.5.3.2

Combine using the product rule for radicals.

$lim_{x \to 2} \frac{\frac{2 \sqrt{x + 7}}{2 \sqrt{(2 x + 5) (x + 7)}} - \frac{1}{2 \sqrt{x + 7}} \cdot \frac{\sqrt{2 x + 5}}{\sqrt{2 x + 5}}}{1}$

Step 1.5.3.3

Multiply $\frac{1}{2 \sqrt{x + 7}}$ by $\frac{\sqrt{2 x + 5}}{\sqrt{2 x + 5}}$ .

$lim_{x \to 2} \frac{\frac{2 \sqrt{x + 7}}{2 \sqrt{(2 x + 5) (x + 7)}} - \frac{\sqrt{2 x + 5}}{2 \sqrt{x + 7} \sqrt{2 x + 5}}}{1}$

Step 1.5.3.4

Combine using the product rule for radicals.

$lim_{x \to 2} \frac{\frac{2 \sqrt{x + 7}}{2 \sqrt{(2 x + 5) (x + 7)}} - \frac{\sqrt{2 x + 5}}{2 \sqrt{(2 x + 5) (x + 7)}}}{1}$

Step 1.5.4

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{\frac{2 \sqrt{x + 7} - \sqrt{2 x + 5}}{2 \sqrt{(2 x + 5) (x + 7)}}}{1}$

Step 1.6

Divide $\frac{2 \sqrt{x + 7} - \sqrt{2 x + 5}}{2 \sqrt{(2 x + 5) (x + 7)}}$ by $1$ .

$lim_{x \to 2} \frac{2 \sqrt{x + 7} - \sqrt{2 x + 5}}{2 \sqrt{(2 x + 5) (x + 7)}}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{1}{2} lim_{x \to 2} \frac{2 \sqrt{x + 7} - \sqrt{2 x + 5}}{\sqrt{(2 x + 5) (x + 7)}}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 2} 2 \sqrt{x + 7} - \sqrt{2 x + 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 2} 2 \sqrt{x + 7} - lim_{x \to 2} \sqrt{2 x + 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.4

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

$\frac{1}{2} \cdot \frac{2 lim_{x \to 2} \sqrt{x + 7} - lim_{x \to 2} \sqrt{2 x + 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.5

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - lim_{x \to 2} \sqrt{2 x + 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + lim_{x \to 2} 7} - lim_{x \to 2} \sqrt{2 x + 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.7

Evaluate the limit of $7$ which is constant as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - lim_{x \to 2} \sqrt{2 x + 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.8

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{lim_{x \to 2} 2 x + 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{lim_{x \to 2} 2 x + lim_{x \to 2} 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.10

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

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{2 lim_{x \to 2} x + lim_{x \to 2} 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.11

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{2 lim_{x \to 2} x + 5}}{lim_{x \to 2} \sqrt{(2 x + 5) (x + 7)}}$

Step 2.12

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{2 lim_{x \to 2} x + 5}}{\sqrt{lim_{x \to 2} (2 x + 5) (x + 7)}}$

Step 2.13

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{2 lim_{x \to 2} x + 5}}{\sqrt{lim_{x \to 2} 2 x + 5 \cdot lim_{x \to 2} x + 7}}$

Step 2.14

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{2 lim_{x \to 2} x + 5}}{\sqrt{(lim_{x \to 2} 2 x + lim_{x \to 2} 5) \cdot lim_{x \to 2} x + 7}}$

Step 2.15

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

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{2 lim_{x \to 2} x + 5}}{\sqrt{(2 lim_{x \to 2} x + lim_{x \to 2} 5) \cdot lim_{x \to 2} x + 7}}$

Step 2.16

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{2 lim_{x \to 2} x + 5}}{\sqrt{(2 lim_{x \to 2} x + 5) \cdot lim_{x \to 2} x + 7}}$

Step 2.17

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{2 lim_{x \to 2} x + 5}}{\sqrt{(2 lim_{x \to 2} x + 5) \cdot (lim_{x \to 2} x + lim_{x \to 2} 7)}}$

Step 2.18

Evaluate the limit of $7$ which is constant as $x$ approaches $2$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 2} x + 7} - \sqrt{2 lim_{x \to 2} x + 5}}{\sqrt{(2 lim_{x \to 2} x + 5) \cdot (lim_{x \to 2} x + 7)}}$

Step 3

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{2 + 7} - \sqrt{2 lim_{x \to 2} x + 5}}{\sqrt{(2 lim_{x \to 2} x + 5) \cdot (lim_{x \to 2} x + 7)}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{2 + 7} - \sqrt{2 \cdot 2 + 5}}{\sqrt{(2 lim_{x \to 2} x + 5) \cdot (lim_{x \to 2} x + 7)}}$

Step 3.3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{2 + 7} - \sqrt{2 \cdot 2 + 5}}{\sqrt{(2 \cdot 2 + 5) \cdot (lim_{x \to 2} x + 7)}}$

Step 3.4

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{2 + 7} - \sqrt{2 \cdot 2 + 5}}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Add $2$ and $7$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{9} - \sqrt{2 \cdot 2 + 5}}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{1}{2} \cdot \frac{2 \sqrt{3^{2}} - \sqrt{2 \cdot 2 + 5}}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.1.3

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

$\frac{1}{2} \cdot \frac{2 \cdot 3 - \sqrt{2 \cdot 2 + 5}}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.1.4

Multiply $2$ by $3$ .

$\frac{1}{2} \cdot \frac{6 - \sqrt{2 \cdot 2 + 5}}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.1.5

Multiply $2$ by $2$ .

$\frac{1}{2} \cdot \frac{6 - \sqrt{4 + 5}}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.1.6

Add $4$ and $5$ .

$\frac{1}{2} \cdot \frac{6 - \sqrt{9}}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.1.7

Rewrite $9$ as $3^{2}$ .

$\frac{1}{2} \cdot \frac{6 - \sqrt{3^{2}}}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.1.8

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

$\frac{1}{2} \cdot \frac{6 - 1 \cdot 3}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.1.9

Multiply $- 1$ by $3$ .

$\frac{1}{2} \cdot \frac{6 - 3}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.1.10

Subtract $3$ from $6$ .

$\frac{1}{2} \cdot \frac{3}{\sqrt{(2 \cdot 2 + 5) \cdot (2 + 7)}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $2$ .

$\frac{1}{2} \cdot \frac{3}{\sqrt{(4 + 5) (2 + 7)}}$

Step 4.2.2

Add $4$ and $5$ .

$\frac{1}{2} \cdot \frac{3}{\sqrt{9 (2 + 7)}}$

Step 4.2.3

Add $2$ and $7$ .

$\frac{1}{2} \cdot \frac{3}{\sqrt{9 \cdot 9}}$

Step 4.2.4

Multiply $9$ by $9$ .

$\frac{1}{2} \cdot \frac{3}{\sqrt{81}}$

Step 4.2.5

Rewrite $81$ as $9^{2}$ .

$\frac{1}{2} \cdot \frac{3}{\sqrt{9^{2}}}$

Step 4.2.6

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

$\frac{1}{2} \cdot \frac{3}{9}$

Step 4.3

Cancel the common factor of $3$ and $9$ .

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Step 4.3.1

Factor $3$ out of $3$ .

$\frac{1}{2} \cdot \frac{3 (1)}{9}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $3$ out of $9$ .

$\frac{1}{2} \cdot \frac{3 \cdot 1}{3 \cdot 3}$

Step 4.3.2.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{3 \cdot 1}{3 \cdot 3}$

Step 4.3.2.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{1}{3}$

Step 4.4

Multiply $\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 4.4.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$\frac{1}{2 \cdot 3}$

Step 4.4.2

Multiply $2$ by $3$ .

$\frac{1}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$