Calculus Examples

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

Step 1

Evaluate the limit of $\frac{{(3 + h)}^{- 1} - 3^{- 1}}{h}$ which is constant as $x$ approaches $0$ .

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

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

$\frac{\frac{1}{3 + h} - 3^{- 1}}{h}$

Step 2.1.2

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

$\frac{\frac{1}{3 + h} - \frac{1}{3}}{h}$

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

$\frac{\frac{3}{3 (3 + h)} - \frac{3 + h}{3 (3 + h)}}{h}$

Step 2.1.6

Combine the numerators over the common denominator.

$\frac{\frac{3 - (3 + h)}{3 (3 + h)}}{h}$

Step 2.1.7

Rewrite $\frac{3 - (3 + h)}{3 (3 + h)}$ in a factored form.

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Step 2.1.7.1

Apply the distributive property.

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

Step 2.1.7.2

Multiply $- 1$ by $3$ .

$\frac{\frac{3 - 3 - h}{3 (3 + h)}}{h}$

Step 2.1.7.3

Subtract $3$ from $3$ .

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

Step 2.1.7.4

Subtract $h$ from $0$ .

$\frac{\frac{- h}{3 (3 + h)}}{h}$

Step 2.1.8

Move the negative in front of the fraction.

$\frac{- \frac{h}{3 (3 + h)}}{h}$

Step 2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3

Cancel the common factor of $h$ .

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Step 2.3.1

Move the leading negative in $- \frac{h}{3 (3 + h)}$ into the numerator.

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

Step 2.3.2

Factor $h$ out of $- h$ .

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

Step 2.3.3

Cancel the common factor.

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

Step 2.3.4

Rewrite the expression.

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

Step 2.4

Move the negative in front of the fraction.

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