Calculus Examples

$lim_{h \to 0} {(9 + h)}^{- 1} - 9^{- 1}$

Step 1

Evaluate the limit.

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Step 1.1

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

$lim_{h \to 0} {(9 + h)}^{- 1} - lim_{h \to 0} 9^{- 1}$

Step 1.2

Move the exponent $- 1$ from ${(9 + h)}^{- 1}$ outside the limit using the Limits Power Rule.

${(lim_{h \to 0} 9 + h)}^{- 1} - lim_{h \to 0} 9^{- 1}$

Step 1.3

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

${(lim_{h \to 0} 9 + lim_{h \to 0} h)}^{- 1} - lim_{h \to 0} 9^{- 1}$

Step 1.4

Evaluate the limit of $9$ which is constant as $h$ approaches $0$ .

${(9 + lim_{h \to 0} h)}^{- 1} - lim_{h \to 0} 9^{- 1}$

Step 1.5

Evaluate the limit of $9^{- 1}$ which is constant as $h$ approaches $0$ .

${(9 + lim_{h \to 0} h)}^{- 1} - 9^{- 1}$

Step 2

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

${(9 + 0)}^{- 1} - 9^{- 1}$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Add $9$ and $0$ .

$9^{- 1} - 9^{- 1}$

Step 3.1.2

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

$\frac{1}{9} - 9^{- 1}$

Step 3.1.3

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

$\frac{1}{9} - \frac{1}{9}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{1 - 1}{9}$

Step 3.3

Subtract $1$ from $1$ .

$\frac{0}{9}$

Step 3.4

Divide $0$ by $9$ .

$0$