Calculus Examples

$lim_{x \to 153} \frac{x - 153}{\sqrt{x + 16} - 13}$

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.

$\frac{lim_{x \to 153} x - 153}{lim_{x \to 153} \sqrt{x + 16} - 13}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $153$ .

$\frac{lim_{x \to 153} x - lim_{x \to 153} 153}{lim_{x \to 153} \sqrt{x + 16} - 13}$

Step 1.1.2.1.2

Evaluate the limit of $153$ which is constant as $x$ approaches $153$ .

$\frac{lim_{x \to 153} x - 1 \cdot 153}{lim_{x \to 153} \sqrt{x + 16} - 13}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $153$ for $x$ .

$\frac{153 - 1 \cdot 153}{lim_{x \to 153} \sqrt{x + 16} - 13}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $- 1$ by $153$ .

$\frac{153 - 153}{lim_{x \to 153} \sqrt{x + 16} - 13}$

Step 1.1.2.3.2

Subtract $153$ from $153$ .

$\frac{0}{lim_{x \to 153} \sqrt{x + 16} - 13}$

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 $153$ .

$\frac{0}{lim_{x \to 153} \sqrt{x + 16} - lim_{x \to 153} 13}$

Step 1.1.3.1.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{x \to 153} x + 16} - lim_{x \to 153} 13}$

Step 1.1.3.1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $153$ .

$\frac{0}{\sqrt{lim_{x \to 153} x + lim_{x \to 153} 16} - lim_{x \to 153} 13}$

Step 1.1.3.1.4

Evaluate the limit of $16$ which is constant as $x$ approaches $153$ .

$\frac{0}{\sqrt{lim_{x \to 153} x + 16} - lim_{x \to 153} 13}$

Step 1.1.3.1.5

Evaluate the limit of $13$ which is constant as $x$ approaches $153$ .

$\frac{0}{\sqrt{lim_{x \to 153} x + 16} - 1 \cdot 13}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $153$ for $x$ .

$\frac{0}{\sqrt{153 + 16} - 1 \cdot 13}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Add $153$ and $16$ .

$\frac{0}{\sqrt{169} - 1 \cdot 13}$

Step 1.1.3.3.1.2

Rewrite $169$ as $13^{2}$ .

$\frac{0}{\sqrt{13^{2}} - 1 \cdot 13}$

Step 1.1.3.3.1.3

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

$\frac{0}{13 - 1 \cdot 13}$

Step 1.1.3.3.1.4

Multiply $- 1$ by $13$ .

$\frac{0}{13 - 13}$

Step 1.1.3.3.2

Subtract $13$ from $13$ .

$\frac{0}{0}$

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.

$lim_{x \to 153} \frac{x - 153}{\sqrt{x + 16} - 13} = lim_{x \to 153} \frac{\frac{d}{d x} [x - 153]}{\frac{d}{d x} [\sqrt{x + 16} - 13]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 153} \frac{\frac{d}{d x} [x - 153]}{\frac{d}{d x} [\sqrt{x + 16} - 13]}$

Step 1.3.2

By the Sum Rule, the derivative of $x - 153$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 153]$ .

$lim_{x \to 153} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [- 153]}{\frac{d}{d x} [\sqrt{x + 16} - 13]}$

Step 1.3.3

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

$lim_{x \to 153} \frac{1 + \frac{d}{d x} [- 153]}{\frac{d}{d x} [\sqrt{x + 16} - 13]}$

Step 1.3.4

Since $- 153$ is constant with respect to $x$ , the derivative of $- 153$ with respect to $x$ is $0$ .

$lim_{x \to 153} \frac{1 + 0}{\frac{d}{d x} [\sqrt{x + 16} - 13]}$

Step 1.3.5

Add $1$ and $0$ .

$lim_{x \to 153} \frac{1}{\frac{d}{d x} [\sqrt{x + 16} - 13]}$

Step 1.3.6

By the Sum Rule, the derivative of $\sqrt{x + 16} - 13$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x + 16}] + \frac{d}{d x} [- 13]$ .

$lim_{x \to 153} \frac{1}{\frac{d}{d x} [\sqrt{x + 16}] + \frac{d}{d x} [- 13]}$

Step 1.3.7

Evaluate $\frac{d}{d x} [\sqrt{x + 16}]$ .

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Step 1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 16}$ as ${(x + 16)}^{\frac{1}{2}}$ .

$lim_{x \to 153} \frac{1}{\frac{d}{d x} [{(x + 16)}^{\frac{1}{2}}] + \frac{d}{d x} [- 13]}$

Step 1.3.7.2

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 $g (x) = x + 16$ .

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Step 1.3.7.2.1

To apply the Chain Rule, set $u$ as $x + 16$ .

$lim_{x \to 153} \frac{1}{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x + 16] + \frac{d}{d x} [- 13]}$

Step 1.3.7.2.2

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

$lim_{x \to 153} \frac{1}{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x + 16] + \frac{d}{d x} [- 13]}$

Step 1.3.7.2.3

Replace all occurrences of $u$ with $x + 16$ .

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 16] + \frac{d}{d x} [- 13]}$

Step 1.3.7.3

By the Sum Rule, the derivative of $x + 16$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [16]$ .

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{\frac{1}{2} - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [16]) + \frac{d}{d x} [- 13]}$

Step 1.3.7.4

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

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{\frac{1}{2} - 1} (1 + \frac{d}{d x} [16]) + \frac{d}{d x} [- 13]}$

Step 1.3.7.5

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{\frac{1}{2} - 1} (1 + 0) + \frac{d}{d x} [- 13]}$

Step 1.3.7.6

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

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0) + \frac{d}{d x} [- 13]}$

Step 1.3.7.7

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

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d x} [- 13]}$

Step 1.3.7.8

Combine the numerators over the common denominator.

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d x} [- 13]}$

Step 1.3.7.9

Simplify the numerator.

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Step 1.3.7.9.1

Multiply $- 1$ by $2$ .

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{\frac{1 - 2}{2}} (1 + 0) + \frac{d}{d x} [- 13]}$

Step 1.3.7.9.2

Subtract $2$ from $1$ .

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{\frac{- 1}{2}} (1 + 0) + \frac{d}{d x} [- 13]}$

Step 1.3.7.10

Move the negative in front of the fraction.

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{- \frac{1}{2}} (1 + 0) + \frac{d}{d x} [- 13]}$

Step 1.3.7.11

Add $1$ and $0$ .

$lim_{x \to 153} \frac{1}{\frac{1}{2} {(x + 16)}^{- \frac{1}{2}} \cdot 1 + \frac{d}{d x} [- 13]}$

Step 1.3.7.12

Combine $\frac{1}{2}$ and ${(x + 16)}^{- \frac{1}{2}}$ .

$lim_{x \to 153} \frac{1}{\frac{{(x + 16)}^{- \frac{1}{2}}}{2} \cdot 1 + \frac{d}{d x} [- 13]}$

Step 1.3.7.13

Multiply $\frac{{(x + 16)}^{- \frac{1}{2}}}{2}$ by $1$ .

$lim_{x \to 153} \frac{1}{\frac{{(x + 16)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 13]}$

Step 1.3.7.14

Move ${(x + 16)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 153} \frac{1}{\frac{1}{2 {(x + 16)}^{\frac{1}{2}}} + \frac{d}{d x} [- 13]}$

Step 1.3.8

Since $- 13$ is constant with respect to $x$ , the derivative of $- 13$ with respect to $x$ is $0$ .

$lim_{x \to 153} \frac{1}{\frac{1}{2 {(x + 16)}^{\frac{1}{2}}} + 0}$

Step 1.3.9

Add $\frac{1}{2 {(x + 16)}^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 153} \frac{1}{\frac{1}{2 {(x + 16)}^{\frac{1}{2}}}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 153} 1 (2 {(x + 16)}^{\frac{1}{2}})$

Step 1.5

Rewrite ${(x + 16)}^{\frac{1}{2}}$ as $\sqrt{x + 16}$ .

$lim_{x \to 153} 1 (2 \sqrt{x + 16})$

Step 1.6

Multiply $2$ by $1$ .

$lim_{x \to 153} 2 \sqrt{x + 16}$

Step 2

Evaluate the limit.

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Step 2.1

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

$2 lim_{x \to 153} \sqrt{x + 16}$

Step 2.2

Move the limit under the radical sign.

$2 \sqrt{lim_{x \to 153} x + 16}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $153$ .

$2 \sqrt{lim_{x \to 153} x + lim_{x \to 153} 16}$

Step 2.4

Evaluate the limit of $16$ which is constant as $x$ approaches $153$ .

$2 \sqrt{lim_{x \to 153} x + 16}$

Step 3

Evaluate the limit of $x$ by plugging in $153$ for $x$ .

$2 \sqrt{153 + 16}$

Step 4

Simplify the answer.

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Step 4.1

Add $153$ and $16$ .

$2 \sqrt{169}$

Step 4.2

Rewrite $169$ as $13^{2}$ .

$2 \sqrt{13^{2}}$

Step 4.3

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

$2 \cdot 13$

Step 4.4

Multiply $2$ by $13$ .

$26$