Calculus Examples

$\frac{lim_{x \to 1} \frac{1}{x + 8} - \frac{1}{9}}{x - 1}$

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 $x$ approaches $1$ .

$\frac{lim_{x \to 1} \frac{1}{x + 8} - lim_{x \to 1} \frac{1}{9}}{x - 1}$

Step 1.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{\frac{lim_{x \to 1} 1}{lim_{x \to 1} x + 8} - lim_{x \to 1} \frac{1}{9}}{x - 1}$

Step 1.3

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

$\frac{\frac{1}{lim_{x \to 1} x + 8} - lim_{x \to 1} \frac{1}{9}}{x - 1}$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{\frac{1}{lim_{x \to 1} x + lim_{x \to 1} 8} - lim_{x \to 1} \frac{1}{9}}{x - 1}$

Step 1.5

Evaluate the limit of $8$ which is constant as $x$ approaches $1$ .

$\frac{\frac{1}{lim_{x \to 1} x + 8} - lim_{x \to 1} \frac{1}{9}}{x - 1}$

Step 1.6

Evaluate the limit of $\frac{1}{9}$ which is constant as $x$ approaches $1$ .

$\frac{\frac{1}{lim_{x \to 1} x + 8} - \frac{1}{9}}{x - 1}$

Step 2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{\frac{1}{1 + 8} - \frac{1}{9}}{x - 1}$

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(1 + 8) \cdot 9$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{1 + 8} - \frac{1}{9}}{x - 1}$ by $\frac{(1 + 8) \cdot 9}{(1 + 8) \cdot 9}$ .

$\frac{(1 + 8) \cdot 9}{(1 + 8) \cdot 9} \cdot \frac{\frac{1}{1 + 8} - \frac{1}{9}}{x - 1}$

Step 3.1.2

Combine.

$\frac{(1 + 8) \cdot 9 (\frac{1}{1 + 8} - \frac{1}{9})}{(1 + 8) \cdot 9 (x - 1)}$

Step 3.2

Apply the distributive property.

$\frac{(1 + 8) \cdot 9 \frac{1}{1 + 8} + (1 + 8) \cdot 9 (- \frac{1}{9})}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $1 + 8$ .

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Step 3.3.1.1

Factor $1 + 8$ out of $(1 + 8) \cdot 9$ .

$\frac{(1 + 8) (9) \frac{1}{1 + 8} + (1 + 8) \cdot 9 (- \frac{1}{9})}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.3.1.2

Cancel the common factor.

$\frac{(1 + 8) \cdot 9 \frac{1}{1 + 8} + (1 + 8) \cdot 9 (- \frac{1}{9})}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.3.1.3

Rewrite the expression.

$\frac{9 + (1 + 8) \cdot 9 (- \frac{1}{9})}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.3.2

Cancel the common factor of $9$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$\frac{9 + (1 + 8) \cdot 9 \frac{- 1}{9}}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.3.2.2

Factor $9$ out of $(1 + 8) \cdot 9$ .

$\frac{9 + 9 \cdot (1 + 8) \frac{- 1}{9}}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.3.2.3

Cancel the common factor.

$\frac{9 + 9 \cdot (1 + 8) \frac{- 1}{9}}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.3.2.4

Rewrite the expression.

$\frac{9 + (1 + 8) \cdot - 1}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Add $1$ and $8$ .

$\frac{9 + 9 \cdot - 1}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.4.2

Multiply $9$ by $- 1$ .

$\frac{9 - 9}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.4.3

Subtract $9$ from $9$ .

$\frac{0}{(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Factor $(1 + 8) \cdot 9$ out of $(1 + 8) \cdot 9 x + (1 + 8) \cdot 9 \cdot - 1$ .

$\frac{0}{(1 + 8) \cdot 9 (x - 1)}$

Step 3.5.2

Add $1$ and $8$ .

$\frac{0}{9 \cdot 9 (x - 1)}$

Step 3.5.3

Multiply $9$ by $9$ .

$\frac{0}{81 (x - 1)}$

Step 3.6

Cancel the common factor of $0$ and $81$ .

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Step 3.6.1

Factor $81$ out of $0$ .

$\frac{81 (0)}{81 (x - 1)}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Cancel the common factor.

$\frac{81 \cdot 0}{81 (x - 1)}$

Step 3.6.2.2

Rewrite the expression.

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

Step 3.7

Divide $0$ by $x - 1$ .

$0$