Calculus Examples

$lim_{x \to - 2} \frac{\frac{2}{x - 8} + \frac{x}{3 x - 4}}{\frac{x}{2 x - 16} + \frac{1}{3 x - 4}}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{2}{x - 8}$ as a fraction with a common denominator, multiply by $\frac{3 x - 4}{3 x - 4}$ .

$lim_{x \to - 2} \frac{\frac{2}{x - 8} \cdot \frac{3 x - 4}{3 x - 4} + \frac{x}{3 x - 4}}{\frac{x}{2 x - 16} + \frac{1}{3 x - 4}}$

Step 1.2

To write $\frac{x}{3 x - 4}$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$lim_{x \to - 2} \frac{\frac{2}{x - 8} \cdot \frac{3 x - 4}{3 x - 4} + \frac{x}{3 x - 4} \cdot \frac{x - 8}{x - 8}}{\frac{x}{2 x - 16} + \frac{1}{3 x - 4}}$

Step 1.3

Write each expression with a common denominator of $(x - 8) (3 x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2}{x - 8}$ by $\frac{3 x - 4}{3 x - 4}$ .

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4)}{(x - 8) (3 x - 4)} + \frac{x}{3 x - 4} \cdot \frac{x - 8}{x - 8}}{\frac{x}{2 x - 16} + \frac{1}{3 x - 4}}$

Step 1.3.2

Multiply $\frac{x}{3 x - 4}$ by $\frac{x - 8}{x - 8}$ .

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4)}{(x - 8) (3 x - 4)} + \frac{x (x - 8)}{(3 x - 4) (x - 8)}}{\frac{x}{2 x - 16} + \frac{1}{3 x - 4}}$

Step 1.3.3

Reorder the factors of $(x - 8) (3 x - 4)$ .

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4)}{(3 x - 4) (x - 8)} + \frac{x (x - 8)}{(3 x - 4) (x - 8)}}{\frac{x}{2 x - 16} + \frac{1}{3 x - 4}}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4) + x (x - 8)}{(3 x - 4) (x - 8)}}{\frac{x}{2 x - 16} + \frac{1}{3 x - 4}}$

Step 1.5

To write $\frac{x}{2 x - 16}$ as a fraction with a common denominator, multiply by $\frac{3 x - 4}{3 x - 4}$ .

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4) + x (x - 8)}{(3 x - 4) (x - 8)}}{\frac{x}{2 x - 16} \cdot \frac{3 x - 4}{3 x - 4} + \frac{1}{3 x - 4}}$

Step 1.6

To write $\frac{1}{3 x - 4}$ as a fraction with a common denominator, multiply by $\frac{2 x - 16}{2 x - 16}$ .

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4) + x (x - 8)}{(3 x - 4) (x - 8)}}{\frac{x}{2 x - 16} \cdot \frac{3 x - 4}{3 x - 4} + \frac{1}{3 x - 4} \cdot \frac{2 x - 16}{2 x - 16}}$

Step 1.7

Write each expression with a common denominator of $(2 x - 16) (3 x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.7.1

Multiply $\frac{x}{2 x - 16}$ by $\frac{3 x - 4}{3 x - 4}$ .

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4) + x (x - 8)}{(3 x - 4) (x - 8)}}{\frac{x (3 x - 4)}{(2 x - 16) (3 x - 4)} + \frac{1}{3 x - 4} \cdot \frac{2 x - 16}{2 x - 16}}$

Step 1.7.2

Multiply $\frac{1}{3 x - 4}$ by $\frac{2 x - 16}{2 x - 16}$ .

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4) + x (x - 8)}{(3 x - 4) (x - 8)}}{\frac{x (3 x - 4)}{(2 x - 16) (3 x - 4)} + \frac{2 x - 16}{(3 x - 4) (2 x - 16)}}$

Step 1.7.3

Reorder the factors of $(3 x - 4) (2 x - 16)$ .

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4) + x (x - 8)}{(3 x - 4) (x - 8)}}{\frac{x (3 x - 4)}{(2 x - 16) (3 x - 4)} + \frac{2 x - 16}{(2 x - 16) (3 x - 4)}}$

Step 1.8

Combine the numerators over the common denominator.

$lim_{x \to - 2} \frac{\frac{2 (3 x - 4) + x (x - 8)}{(3 x - 4) (x - 8)}}{\frac{x (3 x - 4) + 2 x - 16}{(2 x - 16) (3 x - 4)}}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to - 2} \frac{2 (3 x - 4) + x (x - 8)}{(3 x - 4) (x - 8)} \cdot \frac{(2 x - 16) (3 x - 4)}{x (3 x - 4) + 2 x - 16}$

Step 2.2

Multiply $\frac{2 (3 x - 4) + x (x - 8)}{(3 x - 4) (x - 8)}$ by $\frac{(2 x - 16) (3 x - 4)}{x (3 x - 4) + 2 x - 16}$ .

$lim_{x \to - 2} \frac{(2 (3 x - 4) + x (x - 8)) ((2 x - 16) (3 x - 4))}{(3 x - 4) (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 2.3

Cancel the common factor of $3 x - 4$ .

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Step 2.3.1

Cancel the common factor.

$lim_{x \to - 2} \frac{(2 (3 x - 4) + x (x - 8)) ((2 x - 16) (3 x - 4))}{(3 x - 4) (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 2.3.2

Rewrite the expression.

$lim_{x \to - 2} \frac{(2 (3 x - 4) + x (x - 8)) (2 x - 16)}{(x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to - 2} (2 (3 x - 4) + x (x - 8)) (2 x - 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{lim_{x \to - 2} 2 (3 x - 4) + x (x - 8) \cdot lim_{x \to - 2} 2 x - 16}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{(lim_{x \to - 2} 2 (3 x - 4) + lim_{x \to - 2} x (x - 8)) \cdot lim_{x \to - 2} 2 x - 16}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.3

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

$\frac{(2 lim_{x \to - 2} 3 x - 4 + lim_{x \to - 2} x (x - 8)) \cdot lim_{x \to - 2} 2 x - 16}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{(2 (lim_{x \to - 2} 3 x - lim_{x \to - 2} 4) + lim_{x \to - 2} x (x - 8)) \cdot lim_{x \to - 2} 2 x - 16}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{(2 (3 lim_{x \to - 2} x - lim_{x \to - 2} 4) + lim_{x \to - 2} x (x - 8)) \cdot lim_{x \to - 2} 2 x - 16}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.6

Evaluate the limit of $4$ which is constant as $x$ approaches $- 2$ .

$\frac{(2 (3 lim_{x \to - 2} x - 1 \cdot 4) + lim_{x \to - 2} x (x - 8)) \cdot lim_{x \to - 2} 2 x - 16}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.7

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{(2 (3 lim_{x \to - 2} x - 1 \cdot 4) + lim_{x \to - 2} x \cdot lim_{x \to - 2} x - 8) \cdot lim_{x \to - 2} 2 x - 16}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{(2 (3 lim_{x \to - 2} x - 1 \cdot 4) + lim_{x \to - 2} x \cdot (lim_{x \to - 2} x - lim_{x \to - 2} 8)) \cdot lim_{x \to - 2} 2 x - 16}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.9

Evaluate the limit of $8$ which is constant as $x$ approaches $- 2$ .

$\frac{(2 (3 lim_{x \to - 2} x - 1 \cdot 4) + lim_{x \to - 2} x \cdot (lim_{x \to - 2} x - 1 \cdot 8)) \cdot lim_{x \to - 2} 2 x - 16}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.10

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{(2 (3 lim_{x \to - 2} x - 1 \cdot 4) + lim_{x \to - 2} x \cdot (lim_{x \to - 2} x - 1 \cdot 8)) \cdot (lim_{x \to - 2} 2 x - lim_{x \to - 2} 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.11

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

$\frac{(2 (3 lim_{x \to - 2} x - 1 \cdot 4) + lim_{x \to - 2} x \cdot (lim_{x \to - 2} x - 1 \cdot 8)) \cdot (2 lim_{x \to - 2} x - lim_{x \to - 2} 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.12

Evaluate the limit of $16$ which is constant as $x$ approaches $- 2$ .

$\frac{(2 (3 lim_{x \to - 2} x - 1 \cdot 4) + lim_{x \to - 2} x \cdot (lim_{x \to - 2} x - 1 \cdot 8)) \cdot (2 lim_{x \to - 2} x - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.13

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 3.1.2.13.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{(2 (3 \cdot - 2 - 1 \cdot 4) + lim_{x \to - 2} x \cdot (lim_{x \to - 2} x - 1 \cdot 8)) \cdot (2 lim_{x \to - 2} x - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.13.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{(2 (3 \cdot - 2 - 1 \cdot 4) - 2 \cdot (lim_{x \to - 2} x - 1 \cdot 8)) \cdot (2 lim_{x \to - 2} x - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.13.3

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{(2 (3 \cdot - 2 - 1 \cdot 4) - 2 \cdot (- 2 - 1 \cdot 8)) \cdot (2 lim_{x \to - 2} x - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.13.4

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{(2 (3 \cdot - 2 - 1 \cdot 4) - 2 \cdot (- 2 - 1 \cdot 8)) \cdot (2 \cdot - 2 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14

Simplify the answer.

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Step 3.1.2.14.1

Simplify each term.

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Step 3.1.2.14.1.1

Simplify each term.

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Step 3.1.2.14.1.1.1

Multiply $3$ by $- 2$ .

$\frac{(2 (- 6 - 1 \cdot 4) - 2 \cdot (- 2 - 1 \cdot 8)) \cdot (2 \cdot - 2 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.1.1.2

Multiply $- 1$ by $4$ .

$\frac{(2 (- 6 - 4) - 2 \cdot (- 2 - 1 \cdot 8)) \cdot (2 \cdot - 2 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.1.2

Subtract $4$ from $- 6$ .

$\frac{(2 \cdot - 10 - 2 \cdot (- 2 - 1 \cdot 8)) \cdot (2 \cdot - 2 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.1.3

Multiply $2$ by $- 10$ .

$\frac{(- 20 - 2 \cdot (- 2 - 1 \cdot 8)) \cdot (2 \cdot - 2 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.1.4

Multiply $- 1$ by $8$ .

$\frac{(- 20 - 2 \cdot (- 2 - 8)) \cdot (2 \cdot - 2 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.1.5

Subtract $8$ from $- 2$ .

$\frac{(- 20 - 2 \cdot - 10) \cdot (2 \cdot - 2 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.1.6

Multiply $- 2$ by $- 10$ .

$\frac{(- 20 + 20) \cdot (2 \cdot - 2 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.2

Add $- 20$ and $20$ .

$\frac{0 \cdot (2 \cdot - 2 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.3

Simplify each term.

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Step 3.1.2.14.3.1

Multiply $2$ by $- 2$ .

$\frac{0 \cdot (- 4 - 1 \cdot 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.3.2

Multiply $- 1$ by $16$ .

$\frac{0 \cdot (- 4 - 16)}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.4

Subtract $16$ from $- 4$ .

$\frac{0 \cdot - 20}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.2.14.5

Multiply $0$ by $- 20$ .

$\frac{0}{lim_{x \to - 2} (x - 8) (x (3 x - 4) + 2 x - 16)}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{0}{lim_{x \to - 2} x - 8 \cdot lim_{x \to - 2} x (3 x - 4) + 2 x - 16}$

Step 3.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{0}{(lim_{x \to - 2} x - lim_{x \to - 2} 8) \cdot lim_{x \to - 2} x (3 x - 4) + 2 x - 16}$

Step 3.1.3.3

Evaluate the limit of $8$ which is constant as $x$ approaches $- 2$ .

$\frac{0}{(lim_{x \to - 2} x - 1 \cdot 8) \cdot lim_{x \to - 2} x (3 x - 4) + 2 x - 16}$

Step 3.1.3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{0}{(lim_{x \to - 2} x - 1 \cdot 8) \cdot (lim_{x \to - 2} x (3 x - 4) + lim_{x \to - 2} 2 x - lim_{x \to - 2} 16)}$

Step 3.1.3.5

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{0}{(lim_{x \to - 2} x - 1 \cdot 8) \cdot (lim_{x \to - 2} x \cdot lim_{x \to - 2} 3 x - 4 + lim_{x \to - 2} 2 x - lim_{x \to - 2} 16)}$

Step 3.1.3.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{0}{(lim_{x \to - 2} x - 1 \cdot 8) \cdot (lim_{x \to - 2} x \cdot (lim_{x \to - 2} 3 x - lim_{x \to - 2} 4) + lim_{x \to - 2} 2 x - lim_{x \to - 2} 16)}$

Step 3.1.3.7

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{(lim_{x \to - 2} x - 1 \cdot 8) \cdot (lim_{x \to - 2} x \cdot (3 lim_{x \to - 2} x - lim_{x \to - 2} 4) + lim_{x \to - 2} 2 x - lim_{x \to - 2} 16)}$

Step 3.1.3.8

Evaluate the limit of $4$ which is constant as $x$ approaches $- 2$ .

$\frac{0}{(lim_{x \to - 2} x - 1 \cdot 8) \cdot (lim_{x \to - 2} x \cdot (3 lim_{x \to - 2} x - 1 \cdot 4) + lim_{x \to - 2} 2 x - lim_{x \to - 2} 16)}$

Step 3.1.3.9

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

$\frac{0}{(lim_{x \to - 2} x - 1 \cdot 8) \cdot (lim_{x \to - 2} x \cdot (3 lim_{x \to - 2} x - 1 \cdot 4) + 2 lim_{x \to - 2} x - lim_{x \to - 2} 16)}$

Step 3.1.3.10

Evaluate the limit of $16$ which is constant as $x$ approaches $- 2$ .

$\frac{0}{(lim_{x \to - 2} x - 1 \cdot 8) \cdot (lim_{x \to - 2} x \cdot (3 lim_{x \to - 2} x - 1 \cdot 4) + 2 lim_{x \to - 2} x - 1 \cdot 16)}$

Step 3.1.3.11

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 3.1.3.11.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{0}{(- 2 - 1 \cdot 8) \cdot (lim_{x \to - 2} x \cdot (3 lim_{x \to - 2} x - 1 \cdot 4) + 2 lim_{x \to - 2} x - 1 \cdot 16)}$

Step 3.1.3.11.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{0}{(- 2 - 1 \cdot 8) \cdot (- 2 \cdot (3 lim_{x \to - 2} x - 1 \cdot 4) + 2 lim_{x \to - 2} x - 1 \cdot 16)}$

Step 3.1.3.11.3

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{0}{(- 2 - 1 \cdot 8) \cdot (- 2 \cdot (3 \cdot - 2 - 1 \cdot 4) + 2 lim_{x \to - 2} x - 1 \cdot 16)}$

Step 3.1.3.11.4

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{0}{(- 2 - 1 \cdot 8) \cdot (- 2 \cdot (3 \cdot - 2 - 1 \cdot 4) + 2 \cdot - 2 - 1 \cdot 16)}$

Step 3.1.3.12

Simplify the answer.

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Step 3.1.3.12.1

Multiply $- 1$ by $8$ .

$\frac{0}{(- 2 - 8) \cdot (- 2 \cdot (3 \cdot - 2 - 1 \cdot 4) + 2 \cdot - 2 - 1 \cdot 16)}$

Step 3.1.3.12.2

Subtract $8$ from $- 2$ .

$\frac{0}{- 10 \cdot (- 2 \cdot (3 \cdot - 2 - 1 \cdot 4) + 2 \cdot - 2 - 1 \cdot 16)}$

Step 3.1.3.12.3

Simplify each term.

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Step 3.1.3.12.3.1

Simplify each term.

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Step 3.1.3.12.3.1.1

Multiply $3$ by $- 2$ .

$\frac{0}{- 10 \cdot (- 2 \cdot (- 6 - 1 \cdot 4) + 2 \cdot - 2 - 1 \cdot 16)}$

Step 3.1.3.12.3.1.2

Multiply $- 1$ by $4$ .

$\frac{0}{- 10 \cdot (- 2 \cdot (- 6 - 4) + 2 \cdot - 2 - 1 \cdot 16)}$

Step 3.1.3.12.3.2

Subtract $4$ from $- 6$ .

$\frac{0}{- 10 \cdot (- 2 \cdot - 10 + 2 \cdot - 2 - 1 \cdot 16)}$

Step 3.1.3.12.3.3

Multiply $- 2$ by $- 10$ .

$\frac{0}{- 10 \cdot (20 + 2 \cdot - 2 - 1 \cdot 16)}$

Step 3.1.3.12.3.4

Multiply $2$ by $- 2$ .

$\frac{0}{- 10 \cdot (20 - 4 - 1 \cdot 16)}$

Step 3.1.3.12.3.5

Multiply $- 1$ by $16$ .

$\frac{0}{- 10 \cdot (20 - 4 - 16)}$

Step 3.1.3.12.4

Subtract $4$ from $20$ .

$\frac{0}{- 10 \cdot (16 - 16)}$

Step 3.1.3.12.5

Subtract $16$ from $16$ .

$\frac{0}{- 10 \cdot 0}$

Step 3.1.3.12.6

Multiply $- 10$ by $0$ .

$\frac{0}{0}$

Step 3.1.3.12.7

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.1.3.13

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to - 2} \frac{(2 (3 x - 4) + x (x - 8)) (2 x - 16)}{(x - 8) (x (3 x - 4) + 2 x - 16)} = lim_{x \to - 2} \frac{\frac{d}{d x} [(2 (3 x - 4) + x (x - 8)) (2 x - 16)]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 2} \frac{\frac{d}{d x} [(2 (3 x - 4) + x (x - 8)) (2 x - 16)]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Apply the distributive property.

$lim_{x \to - 2} \frac{\frac{d}{d x} [(2 (3 x) + 2 \cdot - 4 + x (x - 8)) (2 x - 16)]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.2.2

Multiply $3$ by $2$ .

$lim_{x \to - 2} \frac{\frac{d}{d x} [(6 x + 2 \cdot - 4 + x (x - 8)) (2 x - 16)]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.2.3

Multiply $2$ by $- 4$ .

$lim_{x \to - 2} \frac{\frac{d}{d x} [(6 x - 8 + x (x - 8)) (2 x - 16)]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.2.4

Apply the distributive property.

$lim_{x \to - 2} \frac{\frac{d}{d x} [(6 x - 8 + x \cdot x + x \cdot - 8) (2 x - 16)]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.2.5

Multiply $x$ by $x$ .

$lim_{x \to - 2} \frac{\frac{d}{d x} [(6 x - 8 + x^{2} + x \cdot - 8) (2 x - 16)]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.2.6

Move $- 8$ to the left of $x$ .

$lim_{x \to - 2} \frac{\frac{d}{d x} [(6 x - 8 + x^{2} - 8 x) (2 x - 16)]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.3

Subtract $8 x$ from $6 x$ .

$lim_{x \to - 2} \frac{\frac{d}{d x} [(- 2 x - 8 + x^{2}) (2 x - 16)]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 2 x - 8 + x^{2}$ and $g (x) = 2 x - 16$ .

$lim_{x \to - 2} \frac{(- 2 x - 8 + x^{2}) \frac{d}{d x} [2 x - 16] + (2 x - 16) \frac{d}{d x} [- 2 x - 8 + x^{2}]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.5

By the Sum Rule, the derivative of $2 x - 16$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 16]$ .

$lim_{x \to - 2} \frac{(- 2 x - 8 + x^{2}) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 16]) + (2 x - 16) \frac{d}{d x} [- 2 x - 8 + x^{2}]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.6

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to - 2} \frac{(- 2 x - 8 + x^{2}) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 16]) + (2 x - 16) \frac{d}{d x} [- 2 x - 8 + x^{2}]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to - 2} \frac{(- 2 x - 8 + x^{2}) (2 \cdot 1 + \frac{d}{d x} [- 16]) + (2 x - 16) \frac{d}{d x} [- 2 x - 8 + x^{2}]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.8

Multiply $2$ by $1$ .

$lim_{x \to - 2} \frac{(- 2 x - 8 + x^{2}) (2 + \frac{d}{d x} [- 16]) + (2 x - 16) \frac{d}{d x} [- 2 x - 8 + x^{2}]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.9

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{(- 2 x - 8 + x^{2}) (2 + 0) + (2 x - 16) \frac{d}{d x} [- 2 x - 8 + x^{2}]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.10

Add $2$ and $0$ .

$lim_{x \to - 2} \frac{(- 2 x - 8 + x^{2}) \cdot 2 + (2 x - 16) \frac{d}{d x} [- 2 x - 8 + x^{2}]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.11

Move $2$ to the left of $- 2 x - 8 + x^{2}$ .

$lim_{x \to - 2} \frac{2 \cdot (- 2 x - 8 + x^{2}) + (2 x - 16) \frac{d}{d x} [- 2 x - 8 + x^{2}]}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.12

By the Sum Rule, the derivative of $- 2 x - 8 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [x^{2}]$ .

$lim_{x \to - 2} \frac{2 (- 2 x - 8 + x^{2}) + (2 x - 16) (\frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.13

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$lim_{x \to - 2} \frac{2 (- 2 x - 8 + x^{2}) + (2 x - 16) (- 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to - 2} \frac{2 (- 2 x - 8 + x^{2}) + (2 x - 16) (- 2 \cdot 1 + \frac{d}{d x} [- 8] + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.15

Multiply $- 2$ by $1$ .

$lim_{x \to - 2} \frac{2 (- 2 x - 8 + x^{2}) + (2 x - 16) (- 2 + \frac{d}{d x} [- 8] + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.16

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{2 (- 2 x - 8 + x^{2}) + (2 x - 16) (- 2 + 0 + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.17

Add $- 2$ and $0$ .

$lim_{x \to - 2} \frac{2 (- 2 x - 8 + x^{2}) + (2 x - 16) (- 2 + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.18

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to - 2} \frac{2 (- 2 x - 8 + x^{2}) + (2 x - 16) (- 2 + 2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19

Simplify.

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Step 3.3.19.1

Apply the distributive property.

$lim_{x \to - 2} \frac{2 (- 2 x) + 2 \cdot - 8 + 2 x^{2} + (2 x - 16) (- 2 + 2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.2

Apply the distributive property.

$lim_{x \to - 2} \frac{2 (- 2 x) + 2 \cdot - 8 + 2 x^{2} + (2 x - 16) \cdot - 2 + (2 x - 16) (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.3

Apply the distributive property.

$lim_{x \to - 2} \frac{2 (- 2 x) + 2 \cdot - 8 + 2 x^{2} + 2 x \cdot - 2 - 16 \cdot - 2 + (2 x - 16) (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.4

Apply the distributive property.

$lim_{x \to - 2} \frac{2 (- 2 x) + 2 \cdot - 8 + 2 x^{2} + 2 x \cdot - 2 - 16 \cdot - 2 + 2 x (2 x) - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5

Combine terms.

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Step 3.3.19.5.1

Multiply $- 2$ by $2$ .

$lim_{x \to - 2} \frac{- 4 x + 2 \cdot - 8 + 2 x^{2} + 2 x \cdot - 2 - 16 \cdot - 2 + 2 x (2 x) - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.2

Multiply $2$ by $- 8$ .

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} + 2 x \cdot - 2 - 16 \cdot - 2 + 2 x (2 x) - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.3

Multiply $- 2$ by $2$ .

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} - 4 x - 16 \cdot - 2 + 2 x (2 x) - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.4

Multiply $- 16$ by $- 2$ .

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} - 4 x + 32 + 2 x (2 x) - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.5

Multiply $2$ by $2$ .

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} - 4 x + 32 + 4 x \cdot x - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.6

Raise $x$ to the power of $1$ .

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} - 4 x + 32 + 4 (x^{1} x) - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.7

Raise $x$ to the power of $1$ .

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} - 4 x + 32 + 4 (x^{1} x^{1}) - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} - 4 x + 32 + 4 x^{1 + 1} - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.9

Add $1$ and $1$ .

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} - 4 x + 32 + 4 x^{2} - 16 (2 x)}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.10

Multiply $2$ by $- 16$ .

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} - 4 x + 32 + 4 x^{2} - 32 x}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.11

Subtract $32 x$ from $- 4 x$ .

$lim_{x \to - 2} \frac{- 4 x - 16 + 2 x^{2} - 36 x + 32 + 4 x^{2}}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.12

Subtract $36 x$ from $- 4 x$ .

$lim_{x \to - 2} \frac{- 40 x - 16 + 2 x^{2} + 32 + 4 x^{2}}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.13

Add $- 16$ and $32$ .

$lim_{x \to - 2} \frac{- 40 x + 2 x^{2} + 16 + 4 x^{2}}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.5.14

Add $2 x^{2}$ and $4 x^{2}$ .

$lim_{x \to - 2} \frac{- 40 x + 6 x^{2} + 16}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.19.6

Reorder terms.

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{\frac{d}{d x} [(x - 8) (x (3 x - 4) + 2 x - 16)]}$

Step 3.3.20

Simplify each term.

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Step 3.3.20.1

Apply the distributive property.

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{\frac{d}{d x} [(x - 8) (x (3 x) + x \cdot - 4 + 2 x - 16)]}$

Step 3.3.20.2

Rewrite using the commutative property of multiplication.

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{\frac{d}{d x} [(x - 8) (3 x \cdot x + x \cdot - 4 + 2 x - 16)]}$

Step 3.3.20.3

Move $- 4$ to the left of $x$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{\frac{d}{d x} [(x - 8) (3 x \cdot x - 4 \cdot x + 2 x - 16)]}$

Step 3.3.20.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.20.4.1

Move $x$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{\frac{d}{d x} [(x - 8) (3 (x \cdot x) - 4 \cdot x + 2 x - 16)]}$

Step 3.3.20.4.2

Multiply $x$ by $x$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{\frac{d}{d x} [(x - 8) (3 x^{2} - 4 \cdot x + 2 x - 16)]}$

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{\frac{d}{d x} [(x - 8) (3 x^{2} - 4 x + 2 x - 16)]}$

Step 3.3.21

Add $- 4 x$ and $2 x$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{\frac{d}{d x} [(x - 8) (3 x^{2} - 2 x - 16)]}$

Step 3.3.22

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 8$ and $g (x) = 3 x^{2} - 2 x - 16$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) \frac{d}{d x} [3 x^{2} - 2 x - 16] + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.23

By the Sum Rule, the derivative of $3 x^{2} - 2 x - 16$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 16]$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 16]) + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.24

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 16]) + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.25

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (3 (2 x) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 16]) + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.26

Multiply $2$ by $3$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 16]) + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.27

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 16]) + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.28

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2 \cdot 1 + \frac{d}{d x} [- 16]) + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.29

Multiply $- 2$ by $1$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2 + \frac{d}{d x} [- 16]) + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.30

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2 + 0) + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.31

Add $6 x - 2$ and $0$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2) + (3 x^{2} - 2 x - 16) \frac{d}{d x} [x - 8]}$

Step 3.3.32

By the Sum Rule, the derivative of $x - 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 8]$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2) + (3 x^{2} - 2 x - 16) (\frac{d}{d x} [x] + \frac{d}{d x} [- 8])}$

Step 3.3.33

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2) + (3 x^{2} - 2 x - 16) (1 + \frac{d}{d x} [- 8])}$

Step 3.3.34

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2) + (3 x^{2} - 2 x - 16) (1 + 0)}$

Step 3.3.35

Add $1$ and $0$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2) + (3 x^{2} - 2 x - 16) \cdot 1}$

Step 3.3.36

Multiply $3 x^{2} - 2 x - 16$ by $1$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x - 2) + 3 x^{2} - 2 x - 16}$

Step 3.3.37

Simplify.

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Step 3.3.37.1

Apply the distributive property.

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{(x - 8) (6 x) + (x - 8) \cdot - 2 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.2

Apply the distributive property.

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{x (6 x) - 8 (6 x) + (x - 8) \cdot - 2 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.3

Apply the distributive property.

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{x (6 x) - 8 (6 x) + x \cdot - 2 - 8 \cdot - 2 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.4

Combine terms.

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Step 3.3.37.4.1

Raise $x$ to the power of $1$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{6 (x^{1} x) - 8 (6 x) + x \cdot - 2 - 8 \cdot - 2 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.4.2

Raise $x$ to the power of $1$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{6 (x^{1} x^{1}) - 8 (6 x) + x \cdot - 2 - 8 \cdot - 2 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{6 x^{1 + 1} - 8 (6 x) + x \cdot - 2 - 8 \cdot - 2 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.4.4

Add $1$ and $1$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{6 x^{2} - 8 (6 x) + x \cdot - 2 - 8 \cdot - 2 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.4.5

Multiply $6$ by $- 8$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{6 x^{2} - 48 x + x \cdot - 2 - 8 \cdot - 2 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.4.6

Move $- 2$ to the left of $x$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{6 x^{2} - 48 x - 2 \cdot x - 8 \cdot - 2 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.4.7

Multiply $- 8$ by $- 2$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{6 x^{2} - 48 x - 2 x + 16 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.4.8

Subtract $2 x$ from $- 48 x$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{6 x^{2} - 50 x + 16 + 3 x^{2} - 2 x - 16}$

Step 3.3.37.4.9

Add $6 x^{2}$ and $3 x^{2}$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{9 x^{2} - 50 x + 16 - 2 x - 16}$

Step 3.3.37.4.10

Subtract $2 x$ from $- 50 x$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{9 x^{2} - 52 x + 16 - 16}$

Step 3.3.37.4.11

Subtract $16$ from $16$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{9 x^{2} - 52 x + 0}$

Step 3.3.37.4.12

Add $9 x^{2} - 52 x$ and $0$ .

$lim_{x \to - 2} \frac{6 x^{2} - 40 x + 16}{9 x^{2} - 52 x}$

Step 4

Evaluate the limit.

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Step 4.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 2$ .

$\frac{lim_{x \to - 2} 6 x^{2} - 40 x + 16}{lim_{x \to - 2} 9 x^{2} - 52 x}$

Step 4.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{lim_{x \to - 2} 6 x^{2} - lim_{x \to - 2} 40 x + lim_{x \to - 2} 16}{lim_{x \to - 2} 9 x^{2} - 52 x}$

Step 4.3

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

$\frac{6 lim_{x \to - 2} x^{2} - lim_{x \to - 2} 40 x + lim_{x \to - 2} 16}{lim_{x \to - 2} 9 x^{2} - 52 x}$

Step 4.4

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

$\frac{6 {(lim_{x \to - 2} x)}^{2} - lim_{x \to - 2} 40 x + lim_{x \to - 2} 16}{lim_{x \to - 2} 9 x^{2} - 52 x}$

Step 4.5

Move the term $40$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 {(lim_{x \to - 2} x)}^{2} - 40 lim_{x \to - 2} x + lim_{x \to - 2} 16}{lim_{x \to - 2} 9 x^{2} - 52 x}$

Step 4.6

Evaluate the limit of $16$ which is constant as $x$ approaches $- 2$ .

$\frac{6 {(lim_{x \to - 2} x)}^{2} - 40 lim_{x \to - 2} x + 16}{lim_{x \to - 2} 9 x^{2} - 52 x}$

Step 4.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{6 {(lim_{x \to - 2} x)}^{2} - 40 lim_{x \to - 2} x + 16}{lim_{x \to - 2} 9 x^{2} - lim_{x \to - 2} 52 x}$

Step 4.8

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

$\frac{6 {(lim_{x \to - 2} x)}^{2} - 40 lim_{x \to - 2} x + 16}{9 lim_{x \to - 2} x^{2} - lim_{x \to - 2} 52 x}$

Step 4.9

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

$\frac{6 {(lim_{x \to - 2} x)}^{2} - 40 lim_{x \to - 2} x + 16}{9 {(lim_{x \to - 2} x)}^{2} - lim_{x \to - 2} 52 x}$

Step 4.10

Move the term $52$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 {(lim_{x \to - 2} x)}^{2} - 40 lim_{x \to - 2} x + 16}{9 {(lim_{x \to - 2} x)}^{2} - 52 lim_{x \to - 2} x}$

Step 5

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{6 {(- 2)}^{2} - 40 lim_{x \to - 2} x + 16}{9 {(lim_{x \to - 2} x)}^{2} - 52 lim_{x \to - 2} x}$

Step 5.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{6 {(- 2)}^{2} - 40 \cdot - 2 + 16}{9 {(lim_{x \to - 2} x)}^{2} - 52 lim_{x \to - 2} x}$

Step 5.3

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{6 {(- 2)}^{2} - 40 \cdot - 2 + 16}{9 {(- 2)}^{2} - 52 lim_{x \to - 2} x}$

Step 5.4

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{6 {(- 2)}^{2} - 40 \cdot - 2 + 16}{9 {(- 2)}^{2} - 52 \cdot - 2}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$\frac{6 \cdot 4 - 40 \cdot - 2 + 16}{9 {(- 2)}^{2} - 52 \cdot - 2}$

Step 6.1.2

Multiply $6$ by $4$ .

$\frac{24 - 40 \cdot - 2 + 16}{9 {(- 2)}^{2} - 52 \cdot - 2}$

Step 6.1.3

Multiply $- 40$ by $- 2$ .

$\frac{24 + 80 + 16}{9 {(- 2)}^{2} - 52 \cdot - 2}$

Step 6.1.4

Add $24$ and $80$ .

$\frac{104 + 16}{9 {(- 2)}^{2} - 52 \cdot - 2}$

Step 6.1.5

Add $104$ and $16$ .

$\frac{120}{9 {(- 2)}^{2} - 52 \cdot - 2}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

$\frac{120}{9 \cdot 4 - 52 \cdot - 2}$

Step 6.2.2

Multiply $9$ by $4$ .

$\frac{120}{36 - 52 \cdot - 2}$

Step 6.2.3

Multiply $- 52$ by $- 2$ .

$\frac{120}{36 + 104}$

Step 6.2.4

Add $36$ and $104$ .

$\frac{120}{140}$

Step 6.3

Cancel the common factor of $120$ and $140$ .

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Step 6.3.1

Factor $20$ out of $120$ .

$\frac{20 (6)}{140}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $20$ out of $140$ .

$\frac{20 \cdot 6}{20 \cdot 7}$

Step 6.3.2.2

Cancel the common factor.

$\frac{20 \cdot 6}{20 \cdot 7}$

Step 6.3.2.3

Rewrite the expression.

$\frac{6}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{6}{7}$

Decimal Form:

$0.\overline{857142}$