Calculus Examples

$lim_{x \to - 8} - \frac{- 8 + \frac{9}{x}}{8 - \frac{6}{x^{2}}}$

Step 1

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

$- lim_{x \to - 8} \frac{- 8 + \frac{9}{x}}{8 - \frac{6}{x^{2}}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$- \frac{lim_{x \to - 8} - 8 + \frac{9}{x}}{lim_{x \to - 8} 8 - \frac{6}{x^{2}}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$- \frac{- lim_{x \to - 8} 8 + lim_{x \to - 8} \frac{9}{x}}{lim_{x \to - 8} 8 - \frac{6}{x^{2}}}$

Step 4

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

$- \frac{- 1 \cdot 8 + lim_{x \to - 8} \frac{9}{x}}{lim_{x \to - 8} 8 - \frac{6}{x^{2}}}$

Step 5

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$- \frac{- 8 + 9 lim_{x \to - 8} \frac{1}{x}}{lim_{x \to - 8} 8 - \frac{6}{x^{2}}}$

Step 6

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$- \frac{- 8 + 9 \frac{lim_{x \to - 8} 1}{lim_{x \to - 8} x}}{lim_{x \to - 8} 8 - \frac{6}{x^{2}}}$

Step 7

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

$- \frac{- 8 + 9 \frac{1}{lim_{x \to - 8} x}}{lim_{x \to - 8} 8 - \frac{6}{x^{2}}}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$- \frac{- 8 + 9 \frac{1}{lim_{x \to - 8} x}}{lim_{x \to - 8} 8 - lim_{x \to - 8} \frac{6}{x^{2}}}$

Step 9

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

$- \frac{- 8 + 9 \frac{1}{lim_{x \to - 8} x}}{8 - lim_{x \to - 8} \frac{6}{x^{2}}}$

Step 10

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$- \frac{- 8 + 9 \frac{1}{lim_{x \to - 8} x}}{8 - 6 lim_{x \to - 8} \frac{1}{x^{2}}}$

Step 11

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$- \frac{- 8 + 9 \frac{1}{lim_{x \to - 8} x}}{8 - 6 \frac{lim_{x \to - 8} 1}{lim_{x \to - 8} x^{2}}}$

Step 12

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

$- \frac{- 8 + 9 \frac{1}{lim_{x \to - 8} x}}{8 - 6 \frac{1}{lim_{x \to - 8} x^{2}}}$

Step 13

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$- \frac{- 8 + 9 \frac{1}{lim_{x \to - 8} x}}{8 - 6 \frac{1}{{(lim_{x \to - 8} x)}^{2}}}$

Step 14

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 14.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$- \frac{- 8 + 9 (\frac{1}{- 8})}{8 - 6 \frac{1}{{(lim_{x \to - 8} x)}^{2}}}$

Step 14.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$- \frac{- 8 + 9 (\frac{1}{- 8})}{8 - 6 \frac{1}{{(- 8)}^{2}}}$

Step 15

Simplify the answer.

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Step 15.1

Raise $- 8$ to the power of $2$ .

$- \frac{- 8 + 9 (\frac{1}{- 8})}{8 - 6 (\frac{1}{64})}$

Step 15.2

Simplify the numerator.

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Step 15.2.1

Move the negative in front of the fraction.

$- \frac{- 8 + 9 (- \frac{1}{8})}{8 - 6 (\frac{1}{64})}$

Step 15.2.2

Multiply $9 (- \frac{1}{8})$ .

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Step 15.2.2.1

Multiply $- 1$ by $9$ .

$- \frac{- 8 - 9 (\frac{1}{8})}{8 - 6 (\frac{1}{64})}$

Step 15.2.2.2

Combine $- 9$ and $\frac{1}{8}$ .

$- \frac{- 8 + \frac{- 9}{8}}{8 - 6 (\frac{1}{64})}$

Step 15.2.3

Move the negative in front of the fraction.

$- \frac{- 8 - \frac{9}{8}}{8 - 6 (\frac{1}{64})}$

Step 15.2.4

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{- 8 \cdot \frac{8}{8} - \frac{9}{8}}{8 - 6 (\frac{1}{64})}$

Step 15.2.5

Combine $- 8$ and $\frac{8}{8}$ .

$- \frac{\frac{- 8 \cdot 8}{8} - \frac{9}{8}}{8 - 6 (\frac{1}{64})}$

Step 15.2.6

Combine the numerators over the common denominator.

$- \frac{\frac{- 8 \cdot 8 - 9}{8}}{8 - 6 (\frac{1}{64})}$

Step 15.2.7

Simplify the numerator.

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Step 15.2.7.1

Multiply $- 8$ by $8$ .

$- \frac{\frac{- 64 - 9}{8}}{8 - 6 (\frac{1}{64})}$

Step 15.2.7.2

Subtract $9$ from $- 64$ .

$- \frac{\frac{- 73}{8}}{8 - 6 (\frac{1}{64})}$

Step 15.2.8

Move the negative in front of the fraction.

$- \frac{- \frac{73}{8}}{8 - 6 (\frac{1}{64})}$

Step 15.3

Simplify the denominator.

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Step 15.3.1

Cancel the common factor of $2$ .

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Step 15.3.1.1

Factor $2$ out of $- 6$ .

$- \frac{- \frac{73}{8}}{8 + 2 (- 3) \frac{1}{64}}$

Step 15.3.1.2

Factor $2$ out of $64$ .

$- \frac{- \frac{73}{8}}{8 + 2 \cdot - 3 \frac{1}{2 \cdot 32}}$

Step 15.3.1.3

Cancel the common factor.

$- \frac{- \frac{73}{8}}{8 + 2 \cdot - 3 \frac{1}{2 \cdot 32}}$

Step 15.3.1.4

Rewrite the expression.

$- \frac{- \frac{73}{8}}{8 - 3 (\frac{1}{32})}$

Step 15.3.2

Combine $- 3$ and $\frac{1}{32}$ .

$- \frac{- \frac{73}{8}}{8 + \frac{- 3}{32}}$

Step 15.3.3

Move the negative in front of the fraction.

$- \frac{- \frac{73}{8}}{8 - \frac{3}{32}}$

Step 15.3.4

To write $8$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$- \frac{- \frac{73}{8}}{8 \cdot \frac{32}{32} - \frac{3}{32}}$

Step 15.3.5

Combine $8$ and $\frac{32}{32}$ .

$- \frac{- \frac{73}{8}}{\frac{8 \cdot 32}{32} - \frac{3}{32}}$

Step 15.3.6

Combine the numerators over the common denominator.

$- \frac{- \frac{73}{8}}{\frac{8 \cdot 32 - 3}{32}}$

Step 15.3.7

Simplify the numerator.

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Step 15.3.7.1

Multiply $8$ by $32$ .

$- \frac{- \frac{73}{8}}{\frac{256 - 3}{32}}$

Step 15.3.7.2

Subtract $3$ from $256$ .

$- \frac{- \frac{73}{8}}{\frac{253}{32}}$

Step 15.4

Multiply the numerator by the reciprocal of the denominator.

$- (- \frac{73}{8} \cdot \frac{32}{253})$

Step 15.5

Cancel the common factor of $8$ .

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Step 15.5.1

Move the leading negative in $- \frac{73}{8}$ into the numerator.

$- (\frac{- 73}{8} \cdot \frac{32}{253})$

Step 15.5.2

Factor $8$ out of $32$ .

$- (\frac{- 73}{8} \cdot \frac{8 (4)}{253})$

Step 15.5.3

Cancel the common factor.

$- (\frac{- 73}{8} \cdot \frac{8 \cdot 4}{253})$

Step 15.5.4

Rewrite the expression.

$- (- 73 (\frac{4}{253}))$

Step 15.6

Combine $- 73$ and $\frac{4}{253}$ .

$- \frac{- 73 \cdot 4}{253}$

Step 15.7

Multiply $- 73$ by $4$ .

$- \frac{- 292}{253}$

Step 15.8

Move the negative in front of the fraction.

$- - \frac{292}{253}$

Step 15.9

Multiply $- - \frac{292}{253}$ .

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Step 15.9.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{292}{253})$

Step 15.9.2

Multiply $\frac{292}{253}$ by $1$ .

$\frac{292}{253}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{292}{253}$

Decimal Form:

$1.15415019 \dots$