Calculus Examples

$lim_{x \to 0} \frac{2 - {(256 - 7 x)}^{\frac{1}{8}}}{{(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to 0} 2 - {(256 - 7 x)}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

$\frac{lim_{x \to 0} 2 - lim_{x \to 0} {(256 - 7 x)}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.1.2

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

$\frac{2 - lim_{x \to 0} {(256 - 7 x)}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.1.3

Move the exponent $\frac{1}{8}$ from ${(256 - 7 x)}^{\frac{1}{8}}$ outside the limit using the Limits Power Rule.

$\frac{2 - {(lim_{x \to 0} 256 - 7 x)}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.1.4

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

$\frac{2 - {(lim_{x \to 0} 256 - lim_{x \to 0} 7 x)}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.1.5

Evaluate the limit of $256$ which is constant as $x$ approaches $0$ .

$\frac{2 - {(256 - lim_{x \to 0} 7 x)}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.1.6

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 - {(256 - 7 lim_{x \to 0} x)}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.2

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

$\frac{2 - {(256 - 7 \cdot 0)}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $- 7$ by $0$ .

$\frac{2 - {(256 + 0)}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.3.1.2

Add $256$ and $0$ .

$\frac{2 - 256^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.3.1.3

Rewrite $256$ as $2^{8}$ .

$\frac{2 - {(2^{8})}^{\frac{1}{8}}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.3.1.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 - 2^{8 (\frac{1}{8})}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.3.1.5

Cancel the common factor of $8$ .

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Step 1.2.3.1.5.1

Cancel the common factor.

$\frac{2 - 2^{8 (\frac{1}{8})}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.3.1.5.2

Rewrite the expression.

$\frac{2 - 2^{1}}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.3.1.6

Evaluate the exponent.

$\frac{2 - 1 \cdot 2}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.3.1.7

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.2.3.2

Subtract $2$ from $2$ .

$\frac{0}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - 2}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

$\frac{0}{lim_{x \to 0} {(5 x + 32)}^{\frac{1}{5}} - lim_{x \to 0} 2}$

Step 1.3.1.2

Move the exponent $\frac{1}{5}$ from ${(5 x + 32)}^{\frac{1}{5}}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 0} 5 x + 32)}^{\frac{1}{5}} - lim_{x \to 0} 2}$

Step 1.3.1.3

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

$\frac{0}{{(lim_{x \to 0} 5 x + lim_{x \to 0} 32)}^{\frac{1}{5}} - lim_{x \to 0} 2}$

Step 1.3.1.4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{{(5 lim_{x \to 0} x + lim_{x \to 0} 32)}^{\frac{1}{5}} - lim_{x \to 0} 2}$

Step 1.3.1.5

Evaluate the limit of $32$ which is constant as $x$ approaches $0$ .

$\frac{0}{{(5 lim_{x \to 0} x + 32)}^{\frac{1}{5}} - lim_{x \to 0} 2}$

Step 1.3.1.6

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

$\frac{0}{{(5 lim_{x \to 0} x + 32)}^{\frac{1}{5}} - 1 \cdot 2}$

Step 1.3.2

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

$\frac{0}{{(5 \cdot 0 + 32)}^{\frac{1}{5}} - 1 \cdot 2}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply $5$ by $0$ .

$\frac{0}{{(0 + 32)}^{\frac{1}{5}} - 1 \cdot 2}$

Step 1.3.3.1.2

Add $0$ and $32$ .

$\frac{0}{32^{\frac{1}{5}} - 1 \cdot 2}$

Step 1.3.3.1.3

Rewrite $32$ as $2^{5}$ .

$\frac{0}{{(2^{5})}^{\frac{1}{5}} - 1 \cdot 2}$

Step 1.3.3.1.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{0}{2^{5 (\frac{1}{5})} - 1 \cdot 2}$

Step 1.3.3.1.5

Cancel the common factor of $5$ .

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Step 1.3.3.1.5.1

Cancel the common factor.

$\frac{0}{2^{5 (\frac{1}{5})} - 1 \cdot 2}$

Step 1.3.3.1.5.2

Rewrite the expression.

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

Step 1.3.3.1.6

Evaluate the exponent.

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

Step 1.3.3.1.7

Multiply $- 1$ by $2$ .

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

Step 1.3.3.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 0} \frac{2 - {(256 - 7 x)}^{\frac{1}{8}}}{{(5 x + 32)}^{\frac{1}{5}} - 2} = lim_{x \to 0} \frac{\frac{d}{d x} [2 - {(256 - 7 x)}^{\frac{1}{8}}]}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [2 - {(256 - 7 x)}^{\frac{1}{8}}]}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.2

By the Sum Rule, the derivative of $2 - {(256 - 7 x)}^{\frac{1}{8}}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- {(256 - 7 x)}^{\frac{1}{8}}]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [2] + \frac{d}{d x} [- {(256 - 7 x)}^{\frac{1}{8}}]}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.3

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

$lim_{x \to 0} \frac{0 + \frac{d}{d x} [- {(256 - 7 x)}^{\frac{1}{8}}]}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4

Evaluate $\frac{d}{d x} [- {(256 - 7 x)}^{\frac{1}{8}}]$ .

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(256 - 7 x)}^{\frac{1}{8}}$ with respect to $x$ is $- \frac{d}{d x} [{(256 - 7 x)}^{\frac{1}{8}}]$ .

$lim_{x \to 0} \frac{0 - \frac{d}{d x} [{(256 - 7 x)}^{\frac{1}{8}}]}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{8}}$ and $g (x) = 256 - 7 x$ .

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Step 3.4.2.1

To apply the Chain Rule, set $u$ as $256 - 7 x$ .

$lim_{x \to 0} \frac{0 - (\frac{d}{d u} [u^{\frac{1}{8}}] \frac{d}{d x} [256 - 7 x])}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{8}$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} u^{\frac{1}{8} - 1} \frac{d}{d x} [256 - 7 x])}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.2.3

Replace all occurrences of $u$ with $256 - 7 x$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{1}{8} - 1} \frac{d}{d x} [256 - 7 x])}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.3

By the Sum Rule, the derivative of $256 - 7 x$ with respect to $x$ is $\frac{d}{d x} [256] + \frac{d}{d x} [- 7 x]$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{1}{8} - 1} (\frac{d}{d x} [256] + \frac{d}{d x} [- 7 x]))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.4

Since $256$ is constant with respect to $x$ , the derivative of $256$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{1}{8} - 1} (0 + \frac{d}{d x} [- 7 x]))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{1}{8} - 1} (0 - 7 \frac{d}{d x} [x]))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.6

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

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{1}{8} - 1} (0 - 7 \cdot 1))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{1}{8} - 1 \cdot \frac{8}{8}} (0 - 7 \cdot 1))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.8

Combine $- 1$ and $\frac{8}{8}$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{1}{8} + \frac{- 1 \cdot 8}{8}} (0 - 7 \cdot 1))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.9

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{1 - 1 \cdot 8}{8}} (0 - 7 \cdot 1))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.10

Simplify the numerator.

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Step 3.4.10.1

Multiply $- 1$ by $8$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{1 - 8}{8}} (0 - 7 \cdot 1))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.10.2

Subtract $8$ from $1$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{\frac{- 7}{8}} (0 - 7 \cdot 1))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.11

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{- \frac{7}{8}} (0 - 7 \cdot 1))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.12

Multiply $- 7$ by $1$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{- \frac{7}{8}} (0 - 7))}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.13

Subtract $7$ from $0$ .

$lim_{x \to 0} \frac{0 - (\frac{1}{8} {(256 - 7 x)}^{- \frac{7}{8}} \cdot - 7)}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.14

Combine $\frac{1}{8}$ and ${(256 - 7 x)}^{- \frac{7}{8}}$ .

$lim_{x \to 0} \frac{0 - (\frac{{(256 - 7 x)}^{- \frac{7}{8}}}{8} \cdot - 7)}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.15

Combine $\frac{{(256 - 7 x)}^{- \frac{7}{8}}}{8}$ and $- 7$ .

$lim_{x \to 0} \frac{0 - \frac{{(256 - 7 x)}^{- \frac{7}{8}} \cdot - 7}{8}}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.16

Move $- 7$ to the left of ${(256 - 7 x)}^{- \frac{7}{8}}$ .

$lim_{x \to 0} \frac{0 - \frac{- 7 {(256 - 7 x)}^{- \frac{7}{8}}}{8}}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.17

Move ${(256 - 7 x)}^{- \frac{7}{8}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{0 - \frac{- 7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.18

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{0 - - \frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.19

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0} \frac{0 + 1 \frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.4.20

Multiply $\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}$ by $1$ .

$lim_{x \to 0} \frac{0 + \frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.5

Add $0$ and $\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}} - 2]}$

Step 3.6

By the Sum Rule, the derivative of ${(5 x + 32)}^{\frac{1}{5}} - 2$ with respect to $x$ is $\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}}] + \frac{d}{d x} [- 2]$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}}] + \frac{d}{d x} [- 2]}$

Step 3.7

Evaluate $\frac{d}{d x} [{(5 x + 32)}^{\frac{1}{5}}]$ .

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Step 3.7.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{5}}$ and $g (x) = 5 x + 32$ .

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Step 3.7.1.1

To apply the Chain Rule, set $u$ as $5 x + 32$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{d}{d u} [u^{\frac{1}{5}}] \frac{d}{d x} [5 x + 32] + \frac{d}{d x} [- 2]}$

Step 3.7.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{5}$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} u^{\frac{1}{5} - 1} \frac{d}{d x} [5 x + 32] + \frac{d}{d x} [- 2]}$

Step 3.7.1.3

Replace all occurrences of $u$ with $5 x + 32$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{1}{5} - 1} \frac{d}{d x} [5 x + 32] + \frac{d}{d x} [- 2]}$

Step 3.7.2

By the Sum Rule, the derivative of $5 x + 32$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [32]$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{1}{5} - 1} (\frac{d}{d x} [5 x] + \frac{d}{d x} [32]) + \frac{d}{d x} [- 2]}$

Step 3.7.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{1}{5} - 1} (5 \frac{d}{d x} [x] + \frac{d}{d x} [32]) + \frac{d}{d x} [- 2]}$

Step 3.7.4

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

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{1}{5} - 1} (5 \cdot 1 + \frac{d}{d x} [32]) + \frac{d}{d x} [- 2]}$

Step 3.7.5

Since $32$ is constant with respect to $x$ , the derivative of $32$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{1}{5} - 1} (5 \cdot 1 + 0) + \frac{d}{d x} [- 2]}$

Step 3.7.6

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

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{1}{5} - 1 \cdot \frac{5}{5}} (5 \cdot 1 + 0) + \frac{d}{d x} [- 2]}$

Step 3.7.7

Combine $- 1$ and $\frac{5}{5}$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}} (5 \cdot 1 + 0) + \frac{d}{d x} [- 2]}$

Step 3.7.8

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{1 - 1 \cdot 5}{5}} (5 \cdot 1 + 0) + \frac{d}{d x} [- 2]}$

Step 3.7.9

Simplify the numerator.

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Step 3.7.9.1

Multiply $- 1$ by $5$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{1 - 5}{5}} (5 \cdot 1 + 0) + \frac{d}{d x} [- 2]}$

Step 3.7.9.2

Subtract $5$ from $1$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{\frac{- 4}{5}} (5 \cdot 1 + 0) + \frac{d}{d x} [- 2]}$

Step 3.7.10

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{- \frac{4}{5}} (5 \cdot 1 + 0) + \frac{d}{d x} [- 2]}$

Step 3.7.11

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{- \frac{4}{5}} (5 + 0) + \frac{d}{d x} [- 2]}$

Step 3.7.12

Add $5$ and $0$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{5} {(5 x + 32)}^{- \frac{4}{5}} \cdot 5 + \frac{d}{d x} [- 2]}$

Step 3.7.13

Combine $\frac{1}{5}$ and ${(5 x + 32)}^{- \frac{4}{5}}$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{{(5 x + 32)}^{- \frac{4}{5}}}{5} \cdot 5 + \frac{d}{d x} [- 2]}$

Step 3.7.14

Combine $\frac{{(5 x + 32)}^{- \frac{4}{5}}}{5}$ and $5$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{{(5 x + 32)}^{- \frac{4}{5}} \cdot 5}{5} + \frac{d}{d x} [- 2]}$

Step 3.7.15

Move $5$ to the left of ${(5 x + 32)}^{- \frac{4}{5}}$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{5 \cdot {(5 x + 32)}^{- \frac{4}{5}}}{5} + \frac{d}{d x} [- 2]}$

Step 3.7.16

Multiply $5$ by ${(5 x + 32)}^{- \frac{4}{5}}$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{5 {(5 x + 32)}^{- \frac{4}{5}}}{5} + \frac{d}{d x} [- 2]}$

Step 3.7.17

Move ${(5 x + 32)}^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{5}{5 {(5 x + 32)}^{\frac{4}{5}}} + \frac{d}{d x} [- 2]}$

Step 3.7.18

Cancel the common factor.

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{5}{5 {(5 x + 32)}^{\frac{4}{5}}} + \frac{d}{d x} [- 2]}$

Step 3.7.19

Rewrite the expression.

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{{(5 x + 32)}^{\frac{4}{5}}} + \frac{d}{d x} [- 2]}$

Step 3.8

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

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{{(5 x + 32)}^{\frac{4}{5}}} + 0}$

Step 3.9

Add $\frac{1}{{(5 x + 32)}^{\frac{4}{5}}}$ and $0$ .

$lim_{x \to 0} \frac{\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}}{\frac{1}{{(5 x + 32)}^{\frac{4}{5}}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}} {(5 x + 32)}^{\frac{4}{5}}$

Step 5

Combine $\frac{7}{8 {(256 - 7 x)}^{\frac{7}{8}}}$ and ${(5 x + 32)}^{\frac{4}{5}}$ .

$lim_{x \to 0} \frac{7 {(5 x + 32)}^{\frac{4}{5}}}{8 {(256 - 7 x)}^{\frac{7}{8}}}$

Step 6

Move the term $\frac{7}{8}$ outside of the limit because it is constant with respect to $x$ .

$\frac{7}{8} lim_{x \to 0} \frac{{(5 x + 32)}^{\frac{4}{5}}}{{(256 - 7 x)}^{\frac{7}{8}}}$

Step 7

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

$\frac{7}{8} \cdot \frac{lim_{x \to 0} {(5 x + 32)}^{\frac{4}{5}}}{lim_{x \to 0} {(256 - 7 x)}^{\frac{7}{8}}}$

Step 8

Move the exponent $\frac{4}{5}$ from ${(5 x + 32)}^{\frac{4}{5}}$ outside the limit using the Limits Power Rule.

$\frac{7}{8} \cdot \frac{{(lim_{x \to 0} 5 x + 32)}^{\frac{4}{5}}}{lim_{x \to 0} {(256 - 7 x)}^{\frac{7}{8}}}$

Step 9

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

$\frac{7}{8} \cdot \frac{{(lim_{x \to 0} 5 x + lim_{x \to 0} 32)}^{\frac{4}{5}}}{lim_{x \to 0} {(256 - 7 x)}^{\frac{7}{8}}}$

Step 10

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{7}{8} \cdot \frac{{(5 lim_{x \to 0} x + lim_{x \to 0} 32)}^{\frac{4}{5}}}{lim_{x \to 0} {(256 - 7 x)}^{\frac{7}{8}}}$

Step 11

Evaluate the limit of $32$ which is constant as $x$ approaches $0$ .

$\frac{7}{8} \cdot \frac{{(5 lim_{x \to 0} x + 32)}^{\frac{4}{5}}}{lim_{x \to 0} {(256 - 7 x)}^{\frac{7}{8}}}$

Step 12

Move the exponent $\frac{7}{8}$ from ${(256 - 7 x)}^{\frac{7}{8}}$ outside the limit using the Limits Power Rule.

$\frac{7}{8} \cdot \frac{{(5 lim_{x \to 0} x + 32)}^{\frac{4}{5}}}{{(lim_{x \to 0} 256 - 7 x)}^{\frac{7}{8}}}$

Step 13

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

$\frac{7}{8} \cdot \frac{{(5 lim_{x \to 0} x + 32)}^{\frac{4}{5}}}{{(lim_{x \to 0} 256 - lim_{x \to 0} 7 x)}^{\frac{7}{8}}}$

Step 14

Evaluate the limit of $256$ which is constant as $x$ approaches $0$ .

$\frac{7}{8} \cdot \frac{{(5 lim_{x \to 0} x + 32)}^{\frac{4}{5}}}{{(256 - lim_{x \to 0} 7 x)}^{\frac{7}{8}}}$

Step 15

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7}{8} \cdot \frac{{(5 lim_{x \to 0} x + 32)}^{\frac{4}{5}}}{{(256 - 7 lim_{x \to 0} x)}^{\frac{7}{8}}}$

Step 16

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

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Step 16.1

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

$\frac{7}{8} \cdot \frac{{(5 \cdot 0 + 32)}^{\frac{4}{5}}}{{(256 - 7 lim_{x \to 0} x)}^{\frac{7}{8}}}$

Step 16.2

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

$\frac{7}{8} \cdot \frac{{(5 \cdot 0 + 32)}^{\frac{4}{5}}}{{(256 - 7 \cdot 0)}^{\frac{7}{8}}}$

Step 17

Simplify the answer.

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Step 17.1

Combine.

$\frac{7 {(5 \cdot 0 + 32)}^{\frac{4}{5}}}{8 {(256 - 7 \cdot 0)}^{\frac{7}{8}}}$

Step 17.2

Simplify the numerator.

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Step 17.2.1

Multiply $5$ by $0$ .

$\frac{7 {(0 + 32)}^{\frac{4}{5}}}{8 {(256 - 7 \cdot 0)}^{\frac{7}{8}}}$

Step 17.2.2

Add $0$ and $32$ .

$\frac{7 \cdot 32^{\frac{4}{5}}}{8 {(256 - 7 \cdot 0)}^{\frac{7}{8}}}$

Step 17.2.3

Rewrite $32$ as $2^{5}$ .

$\frac{7 \cdot {(2^{5})}^{\frac{4}{5}}}{8 {(256 - 7 \cdot 0)}^{\frac{7}{8}}}$

Step 17.2.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{7 \cdot 2^{5 (\frac{4}{5})}}{8 {(256 - 7 \cdot 0)}^{\frac{7}{8}}}$

Step 17.2.5

Cancel the common factor of $5$ .

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Step 17.2.5.1

Cancel the common factor.

$\frac{7 \cdot 2^{5 (\frac{4}{5})}}{8 {(256 - 7 \cdot 0)}^{\frac{7}{8}}}$

Step 17.2.5.2

Rewrite the expression.

$\frac{7 \cdot 2^{4}}{8 {(256 - 7 \cdot 0)}^{\frac{7}{8}}}$

Step 17.2.6

Raise $2$ to the power of $4$ .

$\frac{7 \cdot 16}{8 {(256 - 7 \cdot 0)}^{\frac{7}{8}}}$

Step 17.3

Simplify the denominator.

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Step 17.3.1

Multiply $- 7$ by $0$ .

$\frac{7 \cdot 16}{8 {(256 + 0)}^{\frac{7}{8}}}$

Step 17.3.2

Add $256$ and $0$ .

$\frac{7 \cdot 16}{8 \cdot 256^{\frac{7}{8}}}$

Step 17.3.3

Rewrite $256$ as $2^{8}$ .

$\frac{7 \cdot 16}{8 \cdot {(2^{8})}^{\frac{7}{8}}}$

Step 17.3.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{7 \cdot 16}{8 \cdot 2^{8 (\frac{7}{8})}}$

Step 17.3.5

Cancel the common factor of $8$ .

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Step 17.3.5.1

Cancel the common factor.

$\frac{7 \cdot 16}{8 \cdot 2^{8 (\frac{7}{8})}}$

Step 17.3.5.2

Rewrite the expression.

$\frac{7 \cdot 16}{8 \cdot 2^{7}}$

Step 17.3.6

Raise $2$ to the power of $7$ .

$\frac{7 \cdot 16}{8 \cdot 128}$

Step 17.4

Multiply $7$ by $16$ .

$\frac{112}{8 \cdot 128}$

Step 17.5

Multiply $8$ by $128$ .

$\frac{112}{1024}$

Step 17.6

Cancel the common factor of $112$ and $1024$ .

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Step 17.6.1

Factor $16$ out of $112$ .

$\frac{16 (7)}{1024}$

Step 17.6.2

Cancel the common factors.

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Step 17.6.2.1

Factor $16$ out of $1024$ .

$\frac{16 \cdot 7}{16 \cdot 64}$

Step 17.6.2.2

Cancel the common factor.

$\frac{16 \cdot 7}{16 \cdot 64}$

Step 17.6.2.3

Rewrite the expression.

$\frac{7}{64}$