Calculus Examples

$lim_{x \to - 8} (\frac{9 x + 5}{x - 6}) (\frac{14 x - 14}{- x - 13})$

Step 1

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

$lim_{x \to - 8} \frac{9 x + 5}{x - 6} \cdot lim_{x \to - 8} \frac{14 x - 14}{- x - 13}$

Step 2

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

$\frac{lim_{x \to - 8} 9 x + 5}{lim_{x \to - 8} x - 6} \cdot lim_{x \to - 8} \frac{14 x - 14}{- x - 13}$

Step 3

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

$\frac{lim_{x \to - 8} 9 x + lim_{x \to - 8} 5}{lim_{x \to - 8} x - 6} \cdot lim_{x \to - 8} \frac{14 x - 14}{- x - 13}$

Step 4

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

$\frac{9 lim_{x \to - 8} x + lim_{x \to - 8} 5}{lim_{x \to - 8} x - 6} \cdot lim_{x \to - 8} \frac{14 x - 14}{- x - 13}$

Step 5

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

$\frac{9 lim_{x \to - 8} x + 5}{lim_{x \to - 8} x - 6} \cdot lim_{x \to - 8} \frac{14 x - 14}{- x - 13}$

Step 6

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

$\frac{9 lim_{x \to - 8} x + 5}{lim_{x \to - 8} x - lim_{x \to - 8} 6} \cdot lim_{x \to - 8} \frac{14 x - 14}{- x - 13}$

Step 7

Evaluate the limit of $6$ which is constant as $x$ approaches $- 8$ .

$\frac{9 lim_{x \to - 8} x + 5}{lim_{x \to - 8} x - 1 \cdot 6} \cdot lim_{x \to - 8} \frac{14 x - 14}{- x - 13}$

Step 8

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

$\frac{9 lim_{x \to - 8} x + 5}{lim_{x \to - 8} x - 1 \cdot 6} \cdot \frac{lim_{x \to - 8} 14 x - 14}{lim_{x \to - 8} - x - 13}$

Step 9

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

$\frac{9 lim_{x \to - 8} x + 5}{lim_{x \to - 8} x - 1 \cdot 6} \cdot \frac{lim_{x \to - 8} 14 x - lim_{x \to - 8} 14}{lim_{x \to - 8} - x - 13}$

Step 10

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 lim_{x \to - 8} x + 5}{lim_{x \to - 8} x - 1 \cdot 6} \cdot \frac{14 lim_{x \to - 8} x - lim_{x \to - 8} 14}{lim_{x \to - 8} - x - 13}$

Step 11

Evaluate the limit of $14$ which is constant as $x$ approaches $- 8$ .

$\frac{9 lim_{x \to - 8} x + 5}{lim_{x \to - 8} x - 1 \cdot 6} \cdot \frac{14 lim_{x \to - 8} x - 1 \cdot 14}{lim_{x \to - 8} - x - 13}$

Step 12

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

$\frac{9 lim_{x \to - 8} x + 5}{lim_{x \to - 8} x - 1 \cdot 6} \cdot \frac{14 lim_{x \to - 8} x - 1 \cdot 14}{- lim_{x \to - 8} x - lim_{x \to - 8} 13}$

Step 13

Evaluate the limit of $13$ which is constant as $x$ approaches $- 8$ .

$\frac{9 lim_{x \to - 8} x + 5}{lim_{x \to - 8} x - 1 \cdot 6} \cdot \frac{14 lim_{x \to - 8} x - 1 \cdot 14}{- lim_{x \to - 8} x - 1 \cdot 13}$

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 $x$ .

$\frac{9 \cdot - 8 + 5}{lim_{x \to - 8} x - 1 \cdot 6} \cdot \frac{14 lim_{x \to - 8} x - 1 \cdot 14}{- lim_{x \to - 8} x - 1 \cdot 13}$

Step 14.2

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

$\frac{9 \cdot - 8 + 5}{- 8 - 1 \cdot 6} \cdot \frac{14 lim_{x \to - 8} x - 1 \cdot 14}{- lim_{x \to - 8} x - 1 \cdot 13}$

Step 14.3

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

$\frac{9 \cdot - 8 + 5}{- 8 - 1 \cdot 6} \cdot \frac{14 \cdot - 8 - 1 \cdot 14}{- lim_{x \to - 8} x - 1 \cdot 13}$

Step 14.4

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

$\frac{9 \cdot - 8 + 5}{- 8 - 1 \cdot 6} \cdot \frac{14 \cdot - 8 - 1 \cdot 14}{8 - 1 \cdot 13}$

Step 15

Simplify the answer.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Multiply $9$ by $- 8$ .

$\frac{- 72 + 5}{- 8 - 1 \cdot 6} \cdot \frac{14 \cdot - 8 - 1 \cdot 14}{8 - 1 \cdot 13}$

Step 15.1.2

Add $- 72$ and $5$ .

$\frac{- 67}{- 8 - 1 \cdot 6} \cdot \frac{14 \cdot - 8 - 1 \cdot 14}{8 - 1 \cdot 13}$

Step 15.2

Simplify the denominator.

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Step 15.2.1

Multiply $- 1$ by $6$ .

$\frac{- 67}{- 8 - 6} \cdot \frac{14 \cdot - 8 - 1 \cdot 14}{8 - 1 \cdot 13}$

Step 15.2.2

Subtract $6$ from $- 8$ .

$\frac{- 67}{- 14} \cdot \frac{14 \cdot - 8 - 1 \cdot 14}{8 - 1 \cdot 13}$

Step 15.3

Simplify the numerator.

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Step 15.3.1

Multiply $14$ by $- 8$ .

$\frac{- 67}{- 14} \cdot \frac{- 112 - 1 \cdot 14}{8 - 1 \cdot 13}$

Step 15.3.2

Multiply $- 1$ by $14$ .

$\frac{- 67}{- 14} \cdot \frac{- 112 - 14}{8 - 1 \cdot 13}$

Step 15.3.3

Subtract $14$ from $- 112$ .

$\frac{- 67}{- 14} \cdot \frac{- 126}{8 - 1 \cdot 13}$

Step 15.4

Simplify the denominator.

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Step 15.4.1

Multiply $- 1$ by $13$ .

$\frac{- 67}{- 14} \cdot \frac{- 126}{8 - 13}$

Step 15.4.2

Subtract $13$ from $8$ .

$\frac{- 67}{- 14} \cdot \frac{- 126}{- 5}$

Step 15.5

Cancel the common factor of $14$ .

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Step 15.5.1

Factor $14$ out of $- 14$ .

$\frac{- 67}{14 (- 1)} \cdot \frac{- 126}{- 5}$

Step 15.5.2

Factor $14$ out of $- 126$ .

$\frac{- 67}{14 \cdot - 1} \cdot \frac{14 \cdot - 9}{- 5}$

Step 15.5.3

Cancel the common factor.

$\frac{- 67}{14 \cdot - 1} \cdot \frac{14 \cdot - 9}{- 5}$

Step 15.5.4

Rewrite the expression.

$\frac{- 67}{- 1} \cdot \frac{- 9}{- 5}$

Step 15.6

Multiply $\frac{- 67}{- 1}$ by $\frac{- 9}{- 5}$ .

$\frac{- 67 \cdot - 9}{- - 5}$

Step 15.7

Multiply $- 67$ by $- 9$ .

$\frac{603}{- - 5}$

Step 15.8

Multiply $- 1$ by $- 5$ .

$\frac{603}{5}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{603}{5}$

Decimal Form:

$120.6$