Calculus Examples

$lim_{x \to 8} \frac{\sqrt{12 - x} - 2}{\sqrt{24 - x} - 4}$

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.

$\frac{lim_{x \to 8} \sqrt{12 - x} - 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

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 $8$ .

$\frac{lim_{x \to 8} \sqrt{12 - x} - lim_{x \to 8} 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 12 - x} - lim_{x \to 8} 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{\sqrt{lim_{x \to 8} 12 - lim_{x \to 8} x} - lim_{x \to 8} 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.4

Evaluate the limit of $12$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{12 - lim_{x \to 8} x} - lim_{x \to 8} 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.5

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{12 - lim_{x \to 8} x} - 1 \cdot 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.6

Simplify terms.

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Step 1.1.2.6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\sqrt{12 - 8} - 1 \cdot 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.6.2

Simplify the answer.

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Step 1.1.2.6.2.1

Simplify each term.

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Step 1.1.2.6.2.1.1

Subtract $8$ from $12$ .

$\frac{\sqrt{4} - 1 \cdot 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.6.2.1.2

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

$\frac{\sqrt{2^{2}} - 1 \cdot 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.6.2.1.3

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

$\frac{2 - 1 \cdot 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.6.2.1.4

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{lim_{x \to 8} \sqrt{24 - x} - 4}$

Step 1.1.2.6.2.2

Subtract $2$ from $2$ .

$\frac{0}{lim_{x \to 8} \sqrt{24 - x} - 4}$

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 $8$ .

$\frac{0}{lim_{x \to 8} \sqrt{24 - x} - lim_{x \to 8} 4}$

Step 1.1.3.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{x \to 8} 24 - x} - lim_{x \to 8} 4}$

Step 1.1.3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{0}{\sqrt{lim_{x \to 8} 24 - lim_{x \to 8} x} - lim_{x \to 8} 4}$

Step 1.1.3.4

Evaluate the limit of $24$ which is constant as $x$ approaches $8$ .

$\frac{0}{\sqrt{24 - lim_{x \to 8} x} - lim_{x \to 8} 4}$

Step 1.1.3.5

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{0}{\sqrt{24 - lim_{x \to 8} x} - 1 \cdot 4}$

Step 1.1.3.6

Simplify terms.

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Step 1.1.3.6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{0}{\sqrt{24 - 8} - 1 \cdot 4}$

Step 1.1.3.6.2

Simplify the answer.

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Step 1.1.3.6.2.1

Simplify each term.

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Step 1.1.3.6.2.1.1

Subtract $8$ from $24$ .

$\frac{0}{\sqrt{16} - 1 \cdot 4}$

Step 1.1.3.6.2.1.2

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

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

Step 1.1.3.6.2.1.3

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

$\frac{0}{4 - 1 \cdot 4}$

Step 1.1.3.6.2.1.4

Multiply $- 1$ by $4$ .

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

Step 1.1.3.6.2.2

Subtract $4$ from $4$ .

$\frac{0}{0}$

Step 1.1.3.6.2.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.7

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.

$lim_{x \to 8} \frac{\sqrt{12 - x} - 2}{\sqrt{24 - x} - 4} = lim_{x \to 8} \frac{\frac{d}{d x} [\sqrt{12 - x} - 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 8} \frac{\frac{d}{d x} [\sqrt{12 - x} - 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt{12 - x} - 2$ with respect to $x$ is $\frac{d}{d x} [\sqrt{12 - x}] + \frac{d}{d x} [- 2]$ .

$lim_{x \to 8} \frac{\frac{d}{d x} [\sqrt{12 - x}] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [\sqrt{12 - x}]$ .

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Step 1.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12 - x}$ as ${(12 - x)}^{\frac{1}{2}}$ .

$lim_{x \to 8} \frac{\frac{d}{d x} [{(12 - x)}^{\frac{1}{2}}] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

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 $g (x) = 12 - x$ .

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Step 1.3.3.2.1

To apply the Chain Rule, set $u$ as $12 - x$ .

$lim_{x \to 8} \frac{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [12 - x] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.2.2

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

$lim_{x \to 8} \frac{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [12 - x] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.2.3

Replace all occurrences of $u$ with $12 - x$ .

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} [12 - x] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.3

By the Sum Rule, the derivative of $12 - x$ with respect to $x$ is $\frac{d}{d x} [12] + \frac{d}{d x} [- x]$ .

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{1}{2} - 1} (\frac{d}{d x} [12] + \frac{d}{d x} [- x]) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.4

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [- x]) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.5

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

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{1}{2} - 1} (0 - \frac{d}{d x} [x]) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

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 $n = 1$ .

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{1}{2} - 1} (0 - 1 \cdot 1) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.7

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

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.8

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

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.9

Combine the numerators over the common denominator.

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{1 - 1 \cdot 2}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.10

Simplify the numerator.

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Step 1.3.3.10.1

Multiply $- 1$ by $2$ .

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{1 - 2}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.10.2

Subtract $2$ from $1$ .

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{\frac{- 1}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.11

Move the negative in front of the fraction.

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{- \frac{1}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.12

Multiply $- 1$ by $1$ .

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{- \frac{1}{2}} (0 - 1) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.13

Subtract $1$ from $0$ .

$lim_{x \to 8} \frac{\frac{1}{2} {(12 - x)}^{- \frac{1}{2}} \cdot - 1 + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.14

Combine $\frac{1}{2}$ and ${(12 - x)}^{- \frac{1}{2}}$ .

$lim_{x \to 8} \frac{\frac{{(12 - x)}^{- \frac{1}{2}}}{2} \cdot - 1 + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.15

Combine $\frac{{(12 - x)}^{- \frac{1}{2}}}{2}$ and $- 1$ .

$lim_{x \to 8} \frac{\frac{{(12 - x)}^{- \frac{1}{2}} \cdot - 1}{2} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.16

Move $- 1$ to the left of ${(12 - x)}^{- \frac{1}{2}}$ .

$lim_{x \to 8} \frac{\frac{- 1 \cdot {(12 - x)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.17

Rewrite $- 1 {(12 - x)}^{- \frac{1}{2}}$ as $- {(12 - x)}^{- \frac{1}{2}}$ .

$lim_{x \to 8} \frac{\frac{- {(12 - x)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.18

Move ${(12 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 8} \frac{\frac{- 1}{2 {(12 - x)}^{\frac{1}{2}}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.3.19

Move the negative in front of the fraction.

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.4

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

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}} + 0}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.5

Add $- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{d}{d x} [\sqrt{24 - x} - 4]}$

Step 1.3.6

By the Sum Rule, the derivative of $\sqrt{24 - x} - 4$ with respect to $x$ is $\frac{d}{d x} [\sqrt{24 - x}] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{d}{d x} [\sqrt{24 - x}] + \frac{d}{d x} [- 4]}$

Step 1.3.7

Evaluate $\frac{d}{d x} [\sqrt{24 - x}]$ .

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Step 1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{24 - x}$ as ${(24 - x)}^{\frac{1}{2}}$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{d}{d x} [{(24 - x)}^{\frac{1}{2}}] + \frac{d}{d x} [- 4]}$

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 $g (x) = 24 - x$ .

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Step 1.3.7.2.1

To apply the Chain Rule, set $u$ as $24 - x$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [24 - x] + \frac{d}{d x} [- 4]}$

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

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [24 - x] + \frac{d}{d x} [- 4]}$

Step 1.3.7.2.3

Replace all occurrences of $u$ with $24 - x$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} [24 - x] + \frac{d}{d x} [- 4]}$

Step 1.3.7.3

By the Sum Rule, the derivative of $24 - x$ with respect to $x$ is $\frac{d}{d x} [24] + \frac{d}{d x} [- x]$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{1}{2} - 1} (\frac{d}{d x} [24] + \frac{d}{d x} [- x]) + \frac{d}{d x} [- 4]}$

Step 1.3.7.4

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [- x]) + \frac{d}{d x} [- 4]}$

Step 1.3.7.5

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

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{1}{2} - 1} (0 - \frac{d}{d x} [x]) + \frac{d}{d x} [- 4]}$

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 $n = 1$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{1}{2} - 1} (0 - 1 \cdot 1) + \frac{d}{d x} [- 4]}$

Step 1.3.7.7

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

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 4]}$

Step 1.3.7.8

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

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 4]}$

Step 1.3.7.9

Combine the numerators over the common denominator.

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{1 - 1 \cdot 2}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 4]}$

Step 1.3.7.10

Simplify the numerator.

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Step 1.3.7.10.1

Multiply $- 1$ by $2$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{1 - 2}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 4]}$

Step 1.3.7.10.2

Subtract $2$ from $1$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{\frac{- 1}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 4]}$

Step 1.3.7.11

Move the negative in front of the fraction.

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{- \frac{1}{2}} (0 - 1 \cdot 1) + \frac{d}{d x} [- 4]}$

Step 1.3.7.12

Multiply $- 1$ by $1$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{- \frac{1}{2}} (0 - 1) + \frac{d}{d x} [- 4]}$

Step 1.3.7.13

Subtract $1$ from $0$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{1}{2} {(24 - x)}^{- \frac{1}{2}} \cdot - 1 + \frac{d}{d x} [- 4]}$

Step 1.3.7.14

Combine $\frac{1}{2}$ and ${(24 - x)}^{- \frac{1}{2}}$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{{(24 - x)}^{- \frac{1}{2}}}{2} \cdot - 1 + \frac{d}{d x} [- 4]}$

Step 1.3.7.15

Combine $\frac{{(24 - x)}^{- \frac{1}{2}}}{2}$ and $- 1$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{{(24 - x)}^{- \frac{1}{2}} \cdot - 1}{2} + \frac{d}{d x} [- 4]}$

Step 1.3.7.16

Move $- 1$ to the left of ${(24 - x)}^{- \frac{1}{2}}$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{- 1 \cdot {(24 - x)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 4]}$

Step 1.3.7.17

Rewrite $- 1 {(24 - x)}^{- \frac{1}{2}}$ as $- {(24 - x)}^{- \frac{1}{2}}$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{- {(24 - x)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 4]}$

Step 1.3.7.18

Move ${(24 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{\frac{- 1}{2 {(24 - x)}^{\frac{1}{2}}} + \frac{d}{d x} [- 4]}$

Step 1.3.7.19

Move the negative in front of the fraction.

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{- \frac{1}{2 {(24 - x)}^{\frac{1}{2}}} + \frac{d}{d x} [- 4]}$

Step 1.3.8

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

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{- \frac{1}{2 {(24 - x)}^{\frac{1}{2}}} + 0}$

Step 1.3.9

Add $- \frac{1}{2 {(24 - x)}^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 8} \frac{- \frac{1}{2 {(12 - x)}^{\frac{1}{2}}}}{- \frac{1}{2 {(24 - x)}^{\frac{1}{2}}}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 8} - \frac{1}{2 {(12 - x)}^{\frac{1}{2}}} (- (2 {(24 - x)}^{\frac{1}{2}}))$

Step 1.5

Convert fractional exponents to radicals.

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Step 1.5.1

Rewrite ${(12 - x)}^{\frac{1}{2}}$ as $\sqrt{12 - x}$ .

$lim_{x \to 8} - \frac{1}{2 \sqrt{12 - x}} (- (2 {(24 - x)}^{\frac{1}{2}}))$

Step 1.5.2

Rewrite ${(24 - x)}^{\frac{1}{2}}$ as $\sqrt{24 - x}$ .

$lim_{x \to 8} - \frac{1}{2 \sqrt{12 - x}} (- (2 \sqrt{24 - x}))$

Step 1.6

Combine factors.

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Step 1.6.1

Multiply $2$ by $- 1$ .

$lim_{x \to 8} - \frac{1}{2 \sqrt{12 - x}} (- 2 \sqrt{24 - x})$

Step 1.6.2

Multiply $- 2$ by $- 1$ .

$lim_{x \to 8} 2 \frac{1}{2 \sqrt{12 - x}} \sqrt{24 - x}$

Step 1.6.3

Combine $2$ and $\frac{1}{2 \sqrt{12 - x}}$ .

$lim_{x \to 8} \frac{2}{2 \sqrt{12 - x}} \sqrt{24 - x}$

Step 1.6.4

Combine $\frac{2}{2 \sqrt{12 - x}}$ and $\sqrt{24 - x}$ .

$lim_{x \to 8} \frac{2 \sqrt{24 - x}}{2 \sqrt{12 - x}}$

Step 1.7

Cancel the common factor of $2$ .

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Step 1.7.1

Cancel the common factor.

$lim_{x \to 8} \frac{2 \sqrt{24 - x}}{2 \sqrt{12 - x}}$

Step 1.7.2

Rewrite the expression.

$lim_{x \to 8} \frac{\sqrt{24 - x}}{\sqrt{12 - x}}$

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 $8$ .

$\frac{lim_{x \to 8} \sqrt{24 - x}}{lim_{x \to 8} \sqrt{12 - x}}$

Step 2.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 24 - x}}{lim_{x \to 8} \sqrt{12 - x}}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{\sqrt{lim_{x \to 8} 24 - lim_{x \to 8} x}}{lim_{x \to 8} \sqrt{12 - x}}$

Step 2.4

Evaluate the limit of $24$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{24 - lim_{x \to 8} x}}{lim_{x \to 8} \sqrt{12 - x}}$

Step 2.5

Move the limit under the radical sign.

$\frac{\sqrt{24 - lim_{x \to 8} x}}{\sqrt{lim_{x \to 8} 12 - x}}$

Step 2.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{\sqrt{24 - lim_{x \to 8} x}}{\sqrt{lim_{x \to 8} 12 - lim_{x \to 8} x}}$

Step 2.7

Evaluate the limit of $12$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{24 - lim_{x \to 8} x}}{\sqrt{12 - lim_{x \to 8} x}}$

Step 3

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\sqrt{24 - 8}}{\sqrt{12 - lim_{x \to 8} x}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\sqrt{24 - 8}}{\sqrt{12 - 8}}$

Step 4

Simplify the answer.

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Step 4.1

Combine $\sqrt{24 - 8}$ and $\sqrt{12 - 8}$ into a single radical.

$\sqrt{\frac{24 - 8}{12 - 8}}$

Step 4.2

Subtract $8$ from $24$ .

$\sqrt{\frac{16}{12 - 8}}$

Step 4.3

Subtract $8$ from $12$ .

$\sqrt{\frac{16}{4}}$

Step 4.4

Divide $16$ by $4$ .

$\sqrt{4}$

Step 4.5

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

$\sqrt{2^{2}}$

Step 4.6

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

$2$