Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the denominator.

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Step 11.1.1

Multiply $- 1$ by $1$ .

$\frac{\frac{1}{0 - 1} + \frac{1}{0 + 1}}{x}$

Step 11.1.2

Subtract $1$ from $0$ .

$\frac{\frac{1}{- 1} + \frac{1}{0 + 1}}{x}$

Step 11.2

Add $0$ and $1$ .

$\frac{\frac{1}{- 1} + \frac{1}{1}}{x}$

Step 11.3

Simplify the numerator.

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Step 11.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{\frac{1}{- 1} + \frac{1}{- 1 (- 1)}}{x}$

Step 11.3.2

Move a negative from the denominator of $\frac{1}{- 1 (- 1)}$ to the numerator.

$\frac{\frac{1}{- 1} + \frac{- 1 \cdot 1}{- 1}}{x}$

Step 11.3.3

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

$\frac{\frac{1}{- 1} \cdot \frac{- 1}{- 1} + \frac{- 1 \cdot 1}{- 1}}{x}$

Step 11.3.4

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

$\frac{\frac{1}{- 1} \cdot \frac{- 1}{- 1} + \frac{- 1 \cdot 1}{- 1} \cdot \frac{- 1}{- 1}}{x}$

Step 11.3.5

Write each expression with a common denominator of $1$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.3.5.1

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

$\frac{\frac{- 1}{- - 1} + \frac{- 1 \cdot 1}{- 1} \cdot \frac{- 1}{- 1}}{x}$

Step 11.3.5.2

Multiply $- 1$ by $- 1$ .

$\frac{\frac{- 1}{1} + \frac{- 1 \cdot 1}{- 1} \cdot \frac{- 1}{- 1}}{x}$

Step 11.3.5.3

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

$\frac{\frac{- 1}{1} + \frac{- 1 \cdot - 1}{- - 1}}{x}$

Step 11.3.5.4

Multiply $- 1$ by $- 1$ .

$\frac{\frac{- 1}{1} + \frac{- 1 \cdot - 1}{1}}{x}$

Step 11.3.6

Combine the numerators over the common denominator.

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

Step 11.3.7

Simplify the numerator.

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Step 11.3.7.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{- 1 + 1}{1}}{x}$

Step 11.3.7.2

Add $- 1$ and $1$ .

$\frac{\frac{0}{1}}{x}$

Step 11.3.8

Divide $0$ by $1$ .

$\frac{0}{x}$

Step 11.4

Divide $0$ by $x$ .

$0$