Calculus Examples

$lim_{h \to 0} \frac{{(8 + h)}^{- 1} - 8^{- 1}}{h}$

Step 1

Evaluate the limit.

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Step 1.1

Simplify the limit argument.

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Step 1.1.1

Convert negative exponents to fractions.

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Step 1.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{h \to 0} \frac{\frac{1}{8 + h} - 8^{- 1}}{h}$

Step 1.1.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{h \to 0} \frac{\frac{1}{8 + h} - \frac{1}{8}}{h}$

Step 1.1.2

Combine terms.

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Step 1.1.2.1

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

$lim_{h \to 0} \frac{\frac{1}{8 + h} \cdot \frac{8}{8} - \frac{1}{8}}{h}$

Step 1.1.2.2

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

$lim_{h \to 0} \frac{\frac{1}{8 + h} \cdot \frac{8}{8} - \frac{1}{8} \cdot \frac{8 + h}{8 + h}}{h}$

Step 1.1.2.3

Write each expression with a common denominator of $(8 + h) \cdot 8$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.2.3.1

Multiply $\frac{1}{8 + h}$ by $\frac{8}{8}$ .

$lim_{h \to 0} \frac{\frac{8}{(8 + h) \cdot 8} - \frac{1}{8} \cdot \frac{8 + h}{8 + h}}{h}$

Step 1.1.2.3.2

Multiply $\frac{1}{8}$ by $\frac{8 + h}{8 + h}$ .

$lim_{h \to 0} \frac{\frac{8}{(8 + h) \cdot 8} - \frac{8 + h}{8 (8 + h)}}{h}$

Step 1.1.2.3.3

Reorder the factors of $(8 + h) \cdot 8$ .

$lim_{h \to 0} \frac{\frac{8}{8 (8 + h)} - \frac{8 + h}{8 (8 + h)}}{h}$

Step 1.1.2.4

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{\frac{8 - (8 + h)}{8 (8 + h)}}{h}$

Step 1.2

Simplify the limit argument.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} \frac{8 - (8 + h)}{8 (8 + h)} \cdot \frac{1}{h}$

Step 1.2.2

Multiply $\frac{8 - (8 + h)}{8 (8 + h)}$ by $\frac{1}{h}$ .

$lim_{h \to 0} \frac{8 - (8 + h)}{8 (8 + h) h}$

Step 1.3

Move the term $\frac{1}{8}$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{8} lim_{h \to 0} \frac{8 - (8 + h)}{(8 + h) h}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{1}{8} \cdot \frac{lim_{h \to 0} 8 - (8 + h)}{lim_{h \to 0} (8 + h) h}$

Step 2.1.2

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

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Step 2.1.2.1

Evaluate the limit of $8 - (8 + h)$ by plugging in $0$ for $h$ .

$\frac{1}{8} \cdot \frac{8 - (8 + 0)}{lim_{h \to 0} (8 + h) h}$

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

Add $8$ and $0$ .

$\frac{1}{8} \cdot \frac{8 - 1 \cdot 8}{lim_{h \to 0} (8 + h) h}$

Step 2.1.2.2.2

Multiply $- 1$ by $8$ .

$\frac{1}{8} \cdot \frac{8 - 8}{lim_{h \to 0} (8 + h) h}$

Step 2.1.2.3

Subtract $8$ from $8$ .

$\frac{1}{8} \cdot \frac{0}{lim_{h \to 0} (8 + h) h}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Split the limit using the Product of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{1}{8} \cdot \frac{0}{lim_{h \to 0} 8 + h \cdot lim_{h \to 0} h}$

Step 2.1.3.2

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

$\frac{1}{8} \cdot \frac{0}{(lim_{h \to 0} 8 + lim_{h \to 0} h) \cdot lim_{h \to 0} h}$

Step 2.1.3.3

Evaluate the limit of $8$ which is constant as $h$ approaches $0$ .

$\frac{1}{8} \cdot \frac{0}{(8 + lim_{h \to 0} h) \cdot lim_{h \to 0} h}$

Step 2.1.3.4

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

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Step 2.1.3.4.1

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

$\frac{1}{8} \cdot \frac{0}{(8 + 0) \cdot lim_{h \to 0} h}$

Step 2.1.3.4.2

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

$\frac{1}{8} \cdot \frac{0}{(8 + 0) \cdot 0}$

Step 2.1.3.5

Simplify the answer.

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Step 2.1.3.5.1

Add $8$ and $0$ .

$\frac{1}{8} \cdot \frac{0}{8 \cdot 0}$

Step 2.1.3.5.2

Multiply $8$ by $0$ .

$\frac{1}{8} \cdot \frac{0}{0}$

Step 2.1.3.5.3

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

Undefined

$\frac{1}{8} \cdot \frac{0}{0}$

Step 2.1.3.6

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

Undefined

$\frac{1}{8} \cdot \frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{1}{8} \cdot \frac{0}{0}$

Step 2.2

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.

$lim_{h \to 0} \frac{8 - (8 + h)}{(8 + h) h} = lim_{h \to 0} \frac{\frac{d}{d h} [8 - (8 + h)]}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$\frac{1}{8} lim_{h \to 0} \frac{\frac{d}{d h} [8 - (8 + h)]}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.2

By the Sum Rule, the derivative of $8 - (8 + h)$ with respect to $h$ is $\frac{d}{d h} [8] + \frac{d}{d h} [- (8 + h)]$ .

$\frac{1}{8} lim_{h \to 0} \frac{\frac{d}{d h} [8] + \frac{d}{d h} [- (8 + h)]}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.3

Since $8$ is constant with respect to $h$ , the derivative of $8$ with respect to $h$ is $0$ .

$\frac{1}{8} lim_{h \to 0} \frac{0 + \frac{d}{d h} [- (8 + h)]}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.4

Evaluate $\frac{d}{d h} [- (8 + h)]$ .

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Step 2.3.4.1

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

$\frac{1}{8} lim_{h \to 0} \frac{0 - \frac{d}{d h} [8 + h]}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.4.2

By the Sum Rule, the derivative of $8 + h$ with respect to $h$ is $\frac{d}{d h} [8] + \frac{d}{d h} [h]$ .

$\frac{1}{8} lim_{h \to 0} \frac{0 - (\frac{d}{d h} [8] + \frac{d}{d h} [h])}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.4.3

Since $8$ is constant with respect to $h$ , the derivative of $8$ with respect to $h$ is $0$ .

$\frac{1}{8} lim_{h \to 0} \frac{0 - (0 + \frac{d}{d h} [h])}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.4.4

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

$\frac{1}{8} lim_{h \to 0} \frac{0 - (0 + 1)}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.4.5

Add $0$ and $1$ .

$\frac{1}{8} lim_{h \to 0} \frac{0 - 1 \cdot 1}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.4.6

Multiply $- 1$ by $1$ .

$\frac{1}{8} lim_{h \to 0} \frac{0 - 1}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.5

Subtract $1$ from $0$ .

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{\frac{d}{d h} [(8 + h) h]}$

Step 2.3.6

Differentiate using the Product Rule which states that $\frac{d}{d h} [f (h) g (h)]$ is $f (h) \frac{d}{d h} [g (h)] + g (h) \frac{d}{d h} [f (h)]$ where $f (h) = 8 + h$ and $g (h) = h$ .

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{(8 + h) \frac{d}{d h} [h] + h \frac{d}{d h} [8 + h]}$

Step 2.3.7

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

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{(8 + h) \cdot 1 + h \frac{d}{d h} [8 + h]}$

Step 2.3.8

Multiply $8 + h$ by $1$ .

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{8 + h + h \frac{d}{d h} [8 + h]}$

Step 2.3.9

By the Sum Rule, the derivative of $8 + h$ with respect to $h$ is $\frac{d}{d h} [8] + \frac{d}{d h} [h]$ .

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{8 + h + h (\frac{d}{d h} [8] + \frac{d}{d h} [h])}$

Step 2.3.10

Since $8$ is constant with respect to $h$ , the derivative of $8$ with respect to $h$ is $0$ .

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{8 + h + h (0 + \frac{d}{d h} [h])}$

Step 2.3.11

Add $0$ and $\frac{d}{d h} [h]$ .

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{8 + h + h \frac{d}{d h} [h]}$

Step 2.3.12

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

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{8 + h + h \cdot 1}$

Step 2.3.13

Multiply $h$ by $1$ .

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{8 + h + h}$

Step 2.3.14

Add $h$ and $h$ .

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{8 + 2 h}$

Step 2.3.15

Reorder terms.

$\frac{1}{8} lim_{h \to 0} \frac{- 1}{2 h + 8}$

Step 3

Evaluate the limit.

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Step 3.1

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

$\frac{1}{8} \cdot \frac{lim_{h \to 0} - 1}{lim_{h \to 0} 2 h + 8}$

Step 3.2

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

$\frac{1}{8} \cdot \frac{- 1}{lim_{h \to 0} 2 h + 8}$

Step 3.3

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

$\frac{1}{8} \cdot \frac{- 1}{lim_{h \to 0} 2 h + lim_{h \to 0} 8}$

Step 3.4

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

$\frac{1}{8} \cdot \frac{- 1}{2 lim_{h \to 0} h + lim_{h \to 0} 8}$

Step 3.5

Evaluate the limit of $8$ which is constant as $h$ approaches $0$ .

$\frac{1}{8} \cdot \frac{- 1}{2 lim_{h \to 0} h + 8}$

Step 4

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

$\frac{1}{8} \cdot \frac{- 1}{2 \cdot 0 + 8}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Multiply $2$ by $0$ .

$\frac{1}{8} \cdot \frac{- 1}{0 + 8}$

Step 5.1.2

Add $0$ and $8$ .

$\frac{1}{8} \cdot \frac{- 1}{8}$

Step 5.2

Move the negative in front of the fraction.

$\frac{1}{8} (- \frac{1}{8})$

Step 5.3

Multiply $\frac{1}{8} (- \frac{1}{8})$ .

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Step 5.3.1

Multiply $\frac{1}{8}$ by $\frac{1}{8}$ .

$- \frac{1}{8 \cdot 8}$

Step 5.3.2

Multiply $8$ by $8$ .

$- \frac{1}{64}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{64}$

Decimal Form:

$- 0.015625$