Calculus Examples

$lim_{x \to 8} \frac{- 5 + \frac{5}{x}}{9 - \frac{2}{x^{2}}}$

Step 1

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

$\frac{lim_{x \to 8} - 5 + \frac{5}{x}}{lim_{x \to 8} 9 - \frac{2}{x^{2}}}$

Step 2

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

$\frac{- lim_{x \to 8} 5 + lim_{x \to 8} \frac{5}{x}}{lim_{x \to 8} 9 - \frac{2}{x^{2}}}$

Step 3

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{- 1 \cdot 5 + lim_{x \to 8} \frac{5}{x}}{lim_{x \to 8} 9 - \frac{2}{x^{2}}}$

Step 4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 5 + 5 lim_{x \to 8} \frac{1}{x}}{lim_{x \to 8} 9 - \frac{2}{x^{2}}}$

Step 5

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

$\frac{- 5 + 5 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x}}{lim_{x \to 8} 9 - \frac{2}{x^{2}}}$

Step 6

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

$\frac{- 5 + 5 \frac{1}{lim_{x \to 8} x}}{lim_{x \to 8} 9 - \frac{2}{x^{2}}}$

Step 7

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

$\frac{- 5 + 5 \frac{1}{lim_{x \to 8} x}}{lim_{x \to 8} 9 - lim_{x \to 8} \frac{2}{x^{2}}}$

Step 8

Evaluate the limit of $9$ which is constant as $x$ approaches $8$ .

$\frac{- 5 + 5 \frac{1}{lim_{x \to 8} x}}{9 - lim_{x \to 8} \frac{2}{x^{2}}}$

Step 9

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 5 + 5 \frac{1}{lim_{x \to 8} x}}{9 - 2 lim_{x \to 8} \frac{1}{x^{2}}}$

Step 10

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

$\frac{- 5 + 5 \frac{1}{lim_{x \to 8} x}}{9 - 2 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x^{2}}}$

Step 11

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

$\frac{- 5 + 5 \frac{1}{lim_{x \to 8} x}}{9 - 2 \frac{1}{lim_{x \to 8} x^{2}}}$

Step 12

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

$\frac{- 5 + 5 \frac{1}{lim_{x \to 8} x}}{9 - 2 \frac{1}{{(lim_{x \to 8} x)}^{2}}}$

Step 13

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

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Step 13.1

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

$\frac{- 5 + 5 (\frac{1}{8})}{9 - 2 \frac{1}{{(lim_{x \to 8} x)}^{2}}}$

Step 13.2

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

$\frac{- 5 + 5 (\frac{1}{8})}{9 - 2 \frac{1}{8^{2}}}$

Step 14

Simplify the answer.

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Step 14.1

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

$\frac{- 5 + 5 (\frac{1}{8})}{9 - 2 (\frac{1}{64})}$

Step 14.2

Simplify the numerator.

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Step 14.2.1

Combine $5$ and $\frac{1}{8}$ .

$\frac{- 5 + \frac{5}{8}}{9 - 2 (\frac{1}{64})}$

Step 14.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{- 5 \cdot \frac{8}{8} + \frac{5}{8}}{9 - 2 (\frac{1}{64})}$

Step 14.2.3

Combine $- 5$ and $\frac{8}{8}$ .

$\frac{\frac{- 5 \cdot 8}{8} + \frac{5}{8}}{9 - 2 (\frac{1}{64})}$

Step 14.2.4

Combine the numerators over the common denominator.

$\frac{\frac{- 5 \cdot 8 + 5}{8}}{9 - 2 (\frac{1}{64})}$

Step 14.2.5

Simplify the numerator.

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Step 14.2.5.1

Multiply $- 5$ by $8$ .

$\frac{\frac{- 40 + 5}{8}}{9 - 2 (\frac{1}{64})}$

Step 14.2.5.2

Add $- 40$ and $5$ .

$\frac{\frac{- 35}{8}}{9 - 2 (\frac{1}{64})}$

Step 14.2.6

Move the negative in front of the fraction.

$\frac{- \frac{35}{8}}{9 - 2 (\frac{1}{64})}$

Step 14.3

Simplify the denominator.

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Step 14.3.1

Cancel the common factor of $2$ .

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Step 14.3.1.1

Factor $2$ out of $- 2$ .

$\frac{- \frac{35}{8}}{9 + 2 (- 1) \frac{1}{64}}$

Step 14.3.1.2

Factor $2$ out of $64$ .

$\frac{- \frac{35}{8}}{9 + 2 \cdot - 1 \frac{1}{2 \cdot 32}}$

Step 14.3.1.3

Cancel the common factor.

$\frac{- \frac{35}{8}}{9 + 2 \cdot - 1 \frac{1}{2 \cdot 32}}$

Step 14.3.1.4

Rewrite the expression.

$\frac{- \frac{35}{8}}{9 - \frac{1}{32}}$

Step 14.3.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$\frac{- \frac{35}{8}}{9 \cdot \frac{32}{32} - \frac{1}{32}}$

Step 14.3.3

Combine $9$ and $\frac{32}{32}$ .

$\frac{- \frac{35}{8}}{\frac{9 \cdot 32}{32} - \frac{1}{32}}$

Step 14.3.4

Combine the numerators over the common denominator.

$\frac{- \frac{35}{8}}{\frac{9 \cdot 32 - 1}{32}}$

Step 14.3.5

Simplify the numerator.

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Step 14.3.5.1

Multiply $9$ by $32$ .

$\frac{- \frac{35}{8}}{\frac{288 - 1}{32}}$

Step 14.3.5.2

Subtract $1$ from $288$ .

$\frac{- \frac{35}{8}}{\frac{287}{32}}$

Step 14.4

Multiply the numerator by the reciprocal of the denominator.

$- \frac{35}{8} \cdot \frac{32}{287}$

Step 14.5

Cancel the common factor of $7$ .

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Step 14.5.1

Move the leading negative in $- \frac{35}{8}$ into the numerator.

$\frac{- 35}{8} \cdot \frac{32}{287}$

Step 14.5.2

Factor $7$ out of $- 35$ .

$\frac{7 (- 5)}{8} \cdot \frac{32}{287}$

Step 14.5.3

Factor $7$ out of $287$ .

$\frac{7 \cdot - 5}{8} \cdot \frac{32}{7 \cdot 41}$

Step 14.5.4

Cancel the common factor.

$\frac{7 \cdot - 5}{8} \cdot \frac{32}{7 \cdot 41}$

Step 14.5.5

Rewrite the expression.

$\frac{- 5}{8} \cdot \frac{32}{41}$

Step 14.6

Cancel the common factor of $8$ .

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Step 14.6.1

Factor $8$ out of $32$ .

$\frac{- 5}{8} \cdot \frac{8 (4)}{41}$

Step 14.6.2

Cancel the common factor.

$\frac{- 5}{8} \cdot \frac{8 \cdot 4}{41}$

Step 14.6.3

Rewrite the expression.

$- 5 (\frac{4}{41})$

Step 14.7

Combine $- 5$ and $\frac{4}{41}$ .

$\frac{- 5 \cdot 4}{41}$

Step 14.8

Multiply $- 5$ by $4$ .

$\frac{- 20}{41}$

Step 14.9

Move the negative in front of the fraction.

$- \frac{20}{41}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{20}{41}$

Decimal Form:

$- 0.\overline{48780}$