Calculus Examples

$lim_{h \to 0} \frac{{(- 3 + h)}^{- 1} + 3^{- 1}}{h}$

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 $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{h \to 0} \frac{\frac{1}{- 3 + h} + 3^{- 1}}{h}$

Step 1.1.1.2

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

$lim_{h \to 0} \frac{\frac{1}{- 3 + h} + \frac{1}{3}}{h}$

Step 1.1.2

Combine terms.

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Step 1.1.2.1

To write $\frac{1}{- 3 + h}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$lim_{h \to 0} \frac{\frac{1}{- 3 + h} \cdot \frac{3}{3} + \frac{1}{3}}{h}$

Step 1.1.2.2

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{- 3 + h}{- 3 + h}$ .

$lim_{h \to 0} \frac{\frac{1}{- 3 + h} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{- 3 + h}{- 3 + h}}{h}$

Step 1.1.2.3

Write each expression with a common denominator of $(- 3 + h) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.2.3.1

Multiply $\frac{1}{- 3 + h}$ by $\frac{3}{3}$ .

$lim_{h \to 0} \frac{\frac{3}{(- 3 + h) \cdot 3} + \frac{1}{3} \cdot \frac{- 3 + h}{- 3 + h}}{h}$

Step 1.1.2.3.2

Multiply $\frac{1}{3}$ by $\frac{- 3 + h}{- 3 + h}$ .

$lim_{h \to 0} \frac{\frac{3}{(- 3 + h) \cdot 3} + \frac{- 3 + h}{3 (- 3 + h)}}{h}$

Step 1.1.2.3.3

Reorder the factors of $(- 3 + h) \cdot 3$ .

$lim_{h \to 0} \frac{\frac{3}{3 (- 3 + h)} + \frac{- 3 + h}{3 (- 3 + h)}}{h}$

Step 1.1.2.4

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{\frac{3 - 3 + h}{3 (- 3 + h)}}{h}$

Step 1.1.2.5

Subtract $3$ from $3$ .

$lim_{h \to 0} \frac{\frac{0 + h}{3 (- 3 + h)}}{h}$

Step 1.1.2.6

Add $0$ and $h$ .

$lim_{h \to 0} \frac{\frac{h}{3 (- 3 + h)}}{h}$

Step 1.2

Simplify the limit argument.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} \frac{h}{3 (- 3 + h)} \cdot \frac{1}{h}$

Step 1.2.2

Multiply $\frac{h}{3 (- 3 + h)}$ by $\frac{1}{h}$ .

$lim_{h \to 0} \frac{h}{3 (- 3 + h) h}$

Step 1.2.3

Cancel the common factor of $h$ .

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Step 1.2.3.1

Cancel the common factor.

$lim_{h \to 0} \frac{h}{3 (- 3 + h) h}$

Step 1.2.3.2

Rewrite the expression.

$lim_{h \to 0} \frac{1}{3 (- 3 + h)}$

Step 1.3

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{3} lim_{h \to 0} \frac{1}{- 3 + h}$

Step 1.4

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

$\frac{1}{3} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} - 3 + h}$

Step 1.5

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

$\frac{1}{3} \cdot \frac{1}{lim_{h \to 0} - 3 + h}$

Step 1.6

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

$\frac{1}{3} \cdot \frac{1}{- lim_{h \to 0} 3 + lim_{h \to 0} h}$

Step 1.7

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

$\frac{1}{3} \cdot \frac{1}{- 1 \cdot 3 + lim_{h \to 0} h}$

Step 2

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

$\frac{1}{3} \cdot \frac{1}{- 1 \cdot 3 + 0}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} \cdot \frac{1}{- 3 + 0}$

Step 3.1.2

Add $- 3$ and $0$ .

$\frac{1}{3} \cdot \frac{1}{- 3}$

Step 3.2

Move the negative in front of the fraction.

$\frac{1}{3} (- \frac{1}{3})$

Step 3.3

Multiply $\frac{1}{3} (- \frac{1}{3})$ .

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Step 3.3.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$- \frac{1}{3 \cdot 3}$

Step 3.3.2

Multiply $3$ by $3$ .

$- \frac{1}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{9}$

Decimal Form:

$- 0.\overline{1}$