Calculus Examples

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

Step 1

Combine fractions using a common denominator.

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Step 1.1

Combine the numerators over the common denominator.

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

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

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

Step 1.2.5.3

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

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

Step 1.2.6

Combine the numerators over the common denominator.

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

Step 1.2.7

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

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Step 1.2.7.1

Apply the distributive property.

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

Step 1.2.7.2

Multiply $- 1$ by $3$ .

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

Step 1.2.7.3

Subtract $3$ from $3$ .

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

Step 1.2.7.4

Subtract $x$ from $0$ .

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

Step 1.2.8

Move the negative in front of the fraction.

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

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

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

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

Step 1.4.2

Factor $x$ out of $- x$ .

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

Step 1.4.3

Cancel the common factor.

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

Step 1.4.4

Rewrite the expression.

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

Step 1.5

Move the negative in front of the fraction.

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Add $3$ and $0$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $3$ by $3$ .

$- \frac{1}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{9}$

Decimal Form:

$- 0.\overline{1}$