Calculus Examples

$lim_{x \to - 6} \frac{\frac{x + 2}{x^{2} - 12} - \frac{1}{x}}{x + 6}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{x + 2}{x^{2} - 12}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to - 6} \frac{\frac{x + 2}{x^{2} - 12} \cdot \frac{x}{x} - \frac{1}{x}}{x + 6}$

Step 1.2

To write $- \frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 12}{x^{2} - 12}$ .

$lim_{x \to - 6} \frac{\frac{x + 2}{x^{2} - 12} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{x^{2} - 12}{x^{2} - 12}}{x + 6}$

Step 1.3

Write each expression with a common denominator of $(x^{2} - 12) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{x + 2}{x^{2} - 12}$ by $\frac{x}{x}$ .

$lim_{x \to - 6} \frac{\frac{(x + 2) x}{(x^{2} - 12) x} - \frac{1}{x} \cdot \frac{x^{2} - 12}{x^{2} - 12}}{x + 6}$

Step 1.3.2

Multiply $\frac{1}{x}$ by $\frac{x^{2} - 12}{x^{2} - 12}$ .

$lim_{x \to - 6} \frac{\frac{(x + 2) x}{(x^{2} - 12) x} - \frac{x^{2} - 12}{x (x^{2} - 12)}}{x + 6}$

Step 1.3.3

Reorder the factors of $(x^{2} - 12) x$ .

$lim_{x \to - 6} \frac{\frac{(x + 2) x}{x (x^{2} - 12)} - \frac{x^{2} - 12}{x (x^{2} - 12)}}{x + 6}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to - 6} \frac{\frac{(x + 2) x - (x^{2} - 12)}{x (x^{2} - 12)}}{x + 6}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to - 6} \frac{(x + 2) x - (x^{2} - 12)}{x (x^{2} - 12)} \cdot \frac{1}{x + 6}$

Step 2.2

Multiply $\frac{(x + 2) x - (x^{2} - 12)}{x (x^{2} - 12)}$ by $\frac{1}{x + 6}$ .

$lim_{x \to - 6} \frac{(x + 2) x - (x^{2} - 12)}{x (x^{2} - 12) (x + 6)}$

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.

$\frac{lim_{x \to - 6} (x + 2) x - (x^{2} - 12)}{lim_{x \to - 6} x (x^{2} - 12) (x + 6)}$

Step 3.1.2

Evaluate the limits by plugging in $- 6$ for all occurrences of $x$ .

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Step 3.1.2.1

Evaluate the limit of $(x + 2) x - (x^{2} - 12)$ by plugging in $- 6$ for $x$ .

$\frac{(- 6 + 2) \cdot - 6 - ({(- 6)}^{2} - 12)}{lim_{x \to - 6} x (x^{2} - 12) (x + 6)}$

Step 3.1.2.2

Simplify each term.

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Step 3.1.2.2.1

Add $- 6$ and $2$ .

$\frac{- 4 \cdot - 6 - ({(- 6)}^{2} - 12)}{lim_{x \to - 6} x (x^{2} - 12) (x + 6)}$

Step 3.1.2.2.2

Multiply $- 4$ by $- 6$ .

$\frac{24 - ({(- 6)}^{2} - 12)}{lim_{x \to - 6} x (x^{2} - 12) (x + 6)}$

Step 3.1.2.2.3

Raise $- 6$ to the power of $2$ .

$\frac{24 - (36 - 12)}{lim_{x \to - 6} x (x^{2} - 12) (x + 6)}$

Step 3.1.2.2.4

Subtract $12$ from $36$ .

$\frac{24 - 1 \cdot 24}{lim_{x \to - 6} x (x^{2} - 12) (x + 6)}$

Step 3.1.2.2.5

Multiply $- 1$ by $24$ .

$\frac{24 - 24}{lim_{x \to - 6} x (x^{2} - 12) (x + 6)}$

Step 3.1.2.3

Subtract $24$ from $24$ .

$\frac{0}{lim_{x \to - 6} x (x^{2} - 12) (x + 6)}$

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 $- 6$ .

$\frac{0}{lim_{x \to - 6} x \cdot lim_{x \to - 6} x^{2} - 12 \cdot lim_{x \to - 6} x + 6}$

Step 3.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 6$ .

$\frac{0}{lim_{x \to - 6} x \cdot (lim_{x \to - 6} x^{2} - lim_{x \to - 6} 12) \cdot lim_{x \to - 6} x + 6}$

Step 3.1.3.3

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

$\frac{0}{lim_{x \to - 6} x \cdot ({(lim_{x \to - 6} x)}^{2} - lim_{x \to - 6} 12) \cdot lim_{x \to - 6} x + 6}$

Step 3.1.3.4

Evaluate the limit of $12$ which is constant as $x$ approaches $- 6$ .

$\frac{0}{lim_{x \to - 6} x \cdot ({(lim_{x \to - 6} x)}^{2} - 1 \cdot 12) \cdot lim_{x \to - 6} x + 6}$

Step 3.1.3.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 6$ .

$\frac{0}{lim_{x \to - 6} x \cdot ({(lim_{x \to - 6} x)}^{2} - 1 \cdot 12) \cdot (lim_{x \to - 6} x + lim_{x \to - 6} 6)}$

Step 3.1.3.6

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

$\frac{0}{lim_{x \to - 6} x \cdot ({(lim_{x \to - 6} x)}^{2} - 1 \cdot 12) \cdot (lim_{x \to - 6} x + 6)}$

Step 3.1.3.7

Evaluate the limits by plugging in $- 6$ for all occurrences of $x$ .

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Step 3.1.3.7.1

Evaluate the limit of $x$ by plugging in $- 6$ for $x$ .

$\frac{0}{- 6 \cdot ({(lim_{x \to - 6} x)}^{2} - 1 \cdot 12) \cdot (lim_{x \to - 6} x + 6)}$

Step 3.1.3.7.2

Evaluate the limit of $x$ by plugging in $- 6$ for $x$ .

$\frac{0}{- 6 \cdot ({(- 6)}^{2} - 1 \cdot 12) \cdot (lim_{x \to - 6} x + 6)}$

Step 3.1.3.7.3

Evaluate the limit of $x$ by plugging in $- 6$ for $x$ .

$\frac{0}{- 6 \cdot ({(- 6)}^{2} - 1 \cdot 12) \cdot (- 6 + 6)}$

Step 3.1.3.8

Simplify the answer.

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Step 3.1.3.8.1

Simplify each term.

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Step 3.1.3.8.1.1

Raise $- 6$ to the power of $2$ .

$\frac{0}{- 6 \cdot (36 - 1 \cdot 12) \cdot (- 6 + 6)}$

Step 3.1.3.8.1.2

Multiply $- 1$ by $12$ .

$\frac{0}{- 6 \cdot (36 - 12) \cdot (- 6 + 6)}$

Step 3.1.3.8.2

Subtract $12$ from $36$ .

$\frac{0}{- 6 \cdot 24 \cdot (- 6 + 6)}$

Step 3.1.3.8.3

Multiply $- 6$ by $24$ .

$\frac{0}{- 144 \cdot (- 6 + 6)}$

Step 3.1.3.8.4

Add $- 6$ and $6$ .

$\frac{0}{- 144 \cdot 0}$

Step 3.1.3.8.5

Multiply $- 144$ by $0$ .

$\frac{0}{0}$

Step 3.1.3.8.6

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

Undefined

$\frac{0}{0}$

Step 3.1.3.9

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.

$lim_{x \to - 6} \frac{(x + 2) x - (x^{2} - 12)}{x (x^{2} - 12) (x + 6)} = lim_{x \to - 6} \frac{\frac{d}{d x} [(x + 2) x - (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 6} \frac{\frac{d}{d x} [(x + 2) x - (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.2

By the Sum Rule, the derivative of $(x + 2) x - (x^{2} - 12)$ with respect to $x$ is $\frac{d}{d x} [(x + 2) x] + \frac{d}{d x} [- (x^{2} - 12)]$ .

$lim_{x \to - 6} \frac{\frac{d}{d x} [(x + 2) x] + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.3

Evaluate $\frac{d}{d x} [(x + 2) x]$ .

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Step 3.3.3.1

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 $f (x) = x + 2$ and $g (x) = x$ .

$lim_{x \to - 6} \frac{(x + 2) \frac{d}{d x} [x] + x \frac{d}{d x} [x + 2] + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.3.2

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

$lim_{x \to - 6} \frac{(x + 2) \cdot 1 + x \frac{d}{d x} [x + 2] + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.3.3

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

$lim_{x \to - 6} \frac{(x + 2) \cdot 1 + x (\frac{d}{d x} [x] + \frac{d}{d x} [2]) + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.3.4

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

$lim_{x \to - 6} \frac{(x + 2) \cdot 1 + x (1 + \frac{d}{d x} [2]) + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$lim_{x \to - 6} \frac{(x + 2) \cdot 1 + x (1 + 0) + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.3.6

Multiply $x + 2$ by $1$ .

$lim_{x \to - 6} \frac{x + 2 + x (1 + 0) + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.3.7

Add $1$ and $0$ .

$lim_{x \to - 6} \frac{x + 2 + x \cdot 1 + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.3.8

Multiply $x$ by $1$ .

$lim_{x \to - 6} \frac{x + 2 + x + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.3.9

Add $x$ and $x$ .

$lim_{x \to - 6} \frac{2 x + 2 + \frac{d}{d x} [- (x^{2} - 12)]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.4

Evaluate $\frac{d}{d x} [- (x^{2} - 12)]$ .

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Step 3.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x^{2} - 12)$ with respect to $x$ is $- \frac{d}{d x} [x^{2} - 12]$ .

$lim_{x \to - 6} \frac{2 x + 2 - \frac{d}{d x} [x^{2} - 12]}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.4.2

By the Sum Rule, the derivative of $x^{2} - 12$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12]$ .

$lim_{x \to - 6} \frac{2 x + 2 - (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12])}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.4.3

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

$lim_{x \to - 6} \frac{2 x + 2 - (2 x + \frac{d}{d x} [- 12])}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.4.4

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$lim_{x \to - 6} \frac{2 x + 2 - (2 x + 0)}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.4.5

Add $2 x$ and $0$ .

$lim_{x \to - 6} \frac{2 x + 2 - (2 x)}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.4.6

Multiply $2$ by $- 1$ .

$lim_{x \to - 6} \frac{2 x + 2 - 2 x}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.5

Combine terms.

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Step 3.3.5.1

Subtract $2 x$ from $2 x$ .

$lim_{x \to - 6} \frac{0 + 2}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.5.2

Add $0$ and $2$ .

$lim_{x \to - 6} \frac{2}{\frac{d}{d x} [x (x^{2} - 12) (x + 6)]}$

Step 3.3.6

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 $f (x) = x (x^{2} - 12)$ and $g (x) = x + 6$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) \frac{d}{d x} [x + 6] + (x + 6) \frac{d}{d x} [x (x^{2} - 12)]}$

Step 3.3.7

By the Sum Rule, the derivative of $x + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) (\frac{d}{d x} [x] + \frac{d}{d x} [6]) + (x + 6) \frac{d}{d x} [x (x^{2} - 12)]}$

Step 3.3.8

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

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) (1 + \frac{d}{d x} [6]) + (x + 6) \frac{d}{d x} [x (x^{2} - 12)]}$

Step 3.3.9

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) (1 + 0) + (x + 6) \frac{d}{d x} [x (x^{2} - 12)]}$

Step 3.3.10

Add $1$ and $0$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) \cdot 1 + (x + 6) \frac{d}{d x} [x (x^{2} - 12)]}$

Step 3.3.11

Multiply $x$ by $1$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) \frac{d}{d x} [x (x^{2} - 12)]}$

Step 3.3.12

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 $f (x) = x$ and $g (x) = x^{2} - 12$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (x \frac{d}{d x} [x^{2} - 12] + (x^{2} - 12) \frac{d}{d x} [x])}$

Step 3.3.13

By the Sum Rule, the derivative of $x^{2} - 12$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12]$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12]) + (x^{2} - 12) \frac{d}{d x} [x])}$

Step 3.3.14

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

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (x (2 x + \frac{d}{d x} [- 12]) + (x^{2} - 12) \frac{d}{d x} [x])}$

Step 3.3.15

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (x (2 x + 0) + (x^{2} - 12) \frac{d}{d x} [x])}$

Step 3.3.16

Add $2 x$ and $0$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (x (2 x) + (x^{2} - 12) \frac{d}{d x} [x])}$

Step 3.3.17

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

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (2 (x^{1} x) + (x^{2} - 12) \frac{d}{d x} [x])}$

Step 3.3.18

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

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (2 (x^{1} x^{1}) + (x^{2} - 12) \frac{d}{d x} [x])}$

Step 3.3.19

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

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (2 x^{1 + 1} + (x^{2} - 12) \frac{d}{d x} [x])}$

Step 3.3.20

Add $1$ and $1$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (2 x^{2} + (x^{2} - 12) \frac{d}{d x} [x])}$

Step 3.3.21

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

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (2 x^{2} + (x^{2} - 12) \cdot 1)}$

Step 3.3.22

Multiply $x^{2} - 12$ by $1$ .

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (2 x^{2} + x^{2} - 12)}$

Step 3.3.23

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

$lim_{x \to - 6} \frac{2}{x (x^{2} - 12) + (x + 6) (3 x^{2} - 12)}$

Step 3.3.24

Simplify.

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Step 3.3.24.1

Apply the distributive property.

$lim_{x \to - 6} \frac{2}{x \cdot x^{2} + x \cdot - 12 + (x + 6) (3 x^{2} - 12)}$

Step 3.3.24.2

Apply the distributive property.

$lim_{x \to - 6} \frac{2}{x \cdot x^{2} + x \cdot - 12 + (x + 6) (3 x^{2}) + (x + 6) \cdot - 12}$

Step 3.3.24.3

Apply the distributive property.

$lim_{x \to - 6} \frac{2}{x \cdot x^{2} + x \cdot - 12 + x (3 x^{2}) + 6 (3 x^{2}) + (x + 6) \cdot - 12}$

Step 3.3.24.4

Apply the distributive property.

$lim_{x \to - 6} \frac{2}{x \cdot x^{2} + x \cdot - 12 + x (3 x^{2}) + 6 (3 x^{2}) + x \cdot - 12 + 6 \cdot - 12}$

Step 3.3.24.5

Combine terms.

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Step 3.3.24.5.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.3.24.5.1.1

Multiply $x$ by $x^{2}$ .

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Step 3.3.24.5.1.1.1

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

$lim_{x \to - 6} \frac{2}{x^{1} x^{2} + x \cdot - 12 + x (3 x^{2}) + 6 (3 x^{2}) + x \cdot - 12 + 6 \cdot - 12}$

Step 3.3.24.5.1.1.2

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

$lim_{x \to - 6} \frac{2}{x^{1 + 2} + x \cdot - 12 + x (3 x^{2}) + 6 (3 x^{2}) + x \cdot - 12 + 6 \cdot - 12}$

Step 3.3.24.5.1.2

Add $1$ and $2$ .

$lim_{x \to - 6} \frac{2}{x^{3} + x \cdot - 12 + x (3 x^{2}) + 6 (3 x^{2}) + x \cdot - 12 + 6 \cdot - 12}$

Step 3.3.24.5.2

Move $- 12$ to the left of $x$ .

$lim_{x \to - 6} \frac{2}{x^{3} - 12 \cdot x + x (3 x^{2}) + 6 (3 x^{2}) + x \cdot - 12 + 6 \cdot - 12}$

Step 3.3.24.5.3

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

$lim_{x \to - 6} \frac{2}{x^{3} - 12 x + 3 (x^{1} x^{2}) + 6 (3 x^{2}) + x \cdot - 12 + 6 \cdot - 12}$

Step 3.3.24.5.4

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

$lim_{x \to - 6} \frac{2}{x^{3} - 12 x + 3 x^{1 + 2} + 6 (3 x^{2}) + x \cdot - 12 + 6 \cdot - 12}$

Step 3.3.24.5.5

Add $1$ and $2$ .

$lim_{x \to - 6} \frac{2}{x^{3} - 12 x + 3 x^{3} + 6 (3 x^{2}) + x \cdot - 12 + 6 \cdot - 12}$

Step 3.3.24.5.6

Multiply $3$ by $6$ .

$lim_{x \to - 6} \frac{2}{x^{3} - 12 x + 3 x^{3} + 18 x^{2} + x \cdot - 12 + 6 \cdot - 12}$

Step 3.3.24.5.7

Move $- 12$ to the left of $x$ .

$lim_{x \to - 6} \frac{2}{x^{3} - 12 x + 3 x^{3} + 18 x^{2} - 12 \cdot x + 6 \cdot - 12}$

Step 3.3.24.5.8

Multiply $6$ by $- 12$ .

$lim_{x \to - 6} \frac{2}{x^{3} - 12 x + 3 x^{3} + 18 x^{2} - 12 x - 72}$

Step 3.3.24.5.9

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

$lim_{x \to - 6} \frac{2}{4 x^{3} - 12 x + 18 x^{2} - 12 x - 72}$

Step 3.3.24.5.10

Subtract $12 x$ from $- 12 x$ .

$lim_{x \to - 6} \frac{2}{4 x^{3} + 18 x^{2} - 24 x - 72}$

Step 3.4

Cancel the common factor of $2$ and $4 x^{3} + 18 x^{2} - 24 x - 72$ .

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Step 3.4.1

Factor $2$ out of $2$ .

$lim_{x \to - 6} \frac{2 (1)}{4 x^{3} + 18 x^{2} - 24 x - 72}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $4 x^{3}$ .

$lim_{x \to - 6} \frac{2 (1)}{2 (2 x^{3}) + 18 x^{2} - 24 x - 72}$

Step 3.4.2.2

Factor $2$ out of $18 x^{2}$ .

$lim_{x \to - 6} \frac{2 (1)}{2 (2 x^{3}) + 2 (9 x^{2}) - 24 x - 72}$

Step 3.4.2.3

Factor $2$ out of $2 (2 x^{3}) + 2 (9 x^{2})$ .

$lim_{x \to - 6} \frac{2 (1)}{2 (2 x^{3} + 9 x^{2}) - 24 x - 72}$

Step 3.4.2.4

Factor $2$ out of $- 24 x$ .

$lim_{x \to - 6} \frac{2 (1)}{2 (2 x^{3} + 9 x^{2}) + 2 (- 12 x) - 72}$

Step 3.4.2.5

Factor $2$ out of $2 (2 x^{3} + 9 x^{2}) + 2 (- 12 x)$ .

$lim_{x \to - 6} \frac{2 (1)}{2 (2 x^{3} + 9 x^{2} - 12 x) - 72}$

Step 3.4.2.6

Factor $2$ out of $- 72$ .

$lim_{x \to - 6} \frac{2 \cdot 1}{2 (2 x^{3} + 9 x^{2} - 12 x) + 2 \cdot - 36}$

Step 3.4.2.7

Factor $2$ out of $2 (2 x^{3} + 9 x^{2} - 12 x) + 2 (- 36)$ .

$lim_{x \to - 6} \frac{2 (1)}{2 (2 x^{3} + 9 x^{2} - 12 x - 36)}$

Step 3.4.2.8

Cancel the common factor.

$lim_{x \to - 6} \frac{2 \cdot 1}{2 (2 x^{3} + 9 x^{2} - 12 x - 36)}$

Step 3.4.2.9

Rewrite the expression.

$lim_{x \to - 6} \frac{1}{2 x^{3} + 9 x^{2} - 12 x - 36}$

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 $- 6$ .

$\frac{lim_{x \to - 6} 1}{lim_{x \to - 6} 2 x^{3} + 9 x^{2} - 12 x - 36}$

Step 4.2

Evaluate the limit of $1$ which is constant as $x$ approaches $- 6$ .

$\frac{1}{lim_{x \to - 6} 2 x^{3} + 9 x^{2} - 12 x - 36}$

Step 4.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 6$ .

$\frac{1}{lim_{x \to - 6} 2 x^{3} + lim_{x \to - 6} 9 x^{2} - lim_{x \to - 6} 12 x - lim_{x \to - 6} 36}$

Step 4.4

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

$\frac{1}{2 lim_{x \to - 6} x^{3} + lim_{x \to - 6} 9 x^{2} - lim_{x \to - 6} 12 x - lim_{x \to - 6} 36}$

Step 4.5

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{1}{2 {(lim_{x \to - 6} x)}^{3} + lim_{x \to - 6} 9 x^{2} - lim_{x \to - 6} 12 x - lim_{x \to - 6} 36}$

Step 4.6

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

$\frac{1}{2 {(lim_{x \to - 6} x)}^{3} + 9 lim_{x \to - 6} x^{2} - lim_{x \to - 6} 12 x - lim_{x \to - 6} 36}$

Step 4.7

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

$\frac{1}{2 {(lim_{x \to - 6} x)}^{3} + 9 {(lim_{x \to - 6} x)}^{2} - lim_{x \to - 6} 12 x - lim_{x \to - 6} 36}$

Step 4.8

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2 {(lim_{x \to - 6} x)}^{3} + 9 {(lim_{x \to - 6} x)}^{2} - 12 lim_{x \to - 6} x - lim_{x \to - 6} 36}$

Step 4.9

Evaluate the limit of $36$ which is constant as $x$ approaches $- 6$ .

$\frac{1}{2 {(lim_{x \to - 6} x)}^{3} + 9 {(lim_{x \to - 6} x)}^{2} - 12 lim_{x \to - 6} x - 1 \cdot 36}$

Step 5

Evaluate the limits by plugging in $- 6$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $- 6$ for $x$ .

$\frac{1}{2 {(- 6)}^{3} + 9 {(lim_{x \to - 6} x)}^{2} - 12 lim_{x \to - 6} x - 1 \cdot 36}$

Step 5.2

Evaluate the limit of $x$ by plugging in $- 6$ for $x$ .

$\frac{1}{2 {(- 6)}^{3} + 9 {(- 6)}^{2} - 12 lim_{x \to - 6} x - 1 \cdot 36}$

Step 5.3

Evaluate the limit of $x$ by plugging in $- 6$ for $x$ .

$\frac{1}{2 {(- 6)}^{3} + 9 {(- 6)}^{2} - 12 \cdot - 6 - 1 \cdot 36}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

Raise $- 6$ to the power of $3$ .

$\frac{1}{2 \cdot - 216 + 9 {(- 6)}^{2} - 12 \cdot - 6 - 1 \cdot 36}$

Step 6.1.2

Multiply $2$ by $- 216$ .

$\frac{1}{- 432 + 9 {(- 6)}^{2} - 12 \cdot - 6 - 1 \cdot 36}$

Step 6.1.3

Raise $- 6$ to the power of $2$ .

$\frac{1}{- 432 + 9 \cdot 36 - 12 \cdot - 6 - 1 \cdot 36}$

Step 6.1.4

Multiply $9$ by $36$ .

$\frac{1}{- 432 + 324 - 12 \cdot - 6 - 1 \cdot 36}$

Step 6.1.5

Multiply $- 12$ by $- 6$ .

$\frac{1}{- 432 + 324 + 72 - 1 \cdot 36}$

Step 6.1.6

Multiply $- 1$ by $36$ .

$\frac{1}{- 432 + 324 + 72 - 36}$

Step 6.1.7

Add $- 432$ and $324$ .

$\frac{1}{- 108 + 72 - 36}$

Step 6.1.8

Add $- 108$ and $72$ .

$\frac{1}{- 36 - 36}$

Step 6.1.9

Subtract $36$ from $- 36$ .

$\frac{1}{- 72}$

Step 6.2

Move the negative in front of the fraction.

$- \frac{1}{72}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{72}$

Decimal Form:

$- 0.013\overline{8}$