Calculus Examples

$lim_{x \to - 5} \frac{\frac{1}{- 3 x - 8} - \frac{1}{7}}{\frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{1}{- 3 x - 8}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$lim_{x \to - 5} \frac{\frac{1}{- 3 x - 8} \cdot \frac{7}{7} - \frac{1}{7}}{\frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 1.2

To write $- \frac{1}{7}$ as a fraction with a common denominator, multiply by $\frac{- 3 x - 8}{- 3 x - 8}$ .

$lim_{x \to - 5} \frac{\frac{1}{- 3 x - 8} \cdot \frac{7}{7} - \frac{1}{7} \cdot \frac{- 3 x - 8}{- 3 x - 8}}{\frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 1.3

Write each expression with a common denominator of $(- 3 x - 8) \cdot 7$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{- 3 x - 8}$ by $\frac{7}{7}$ .

$lim_{x \to - 5} \frac{\frac{7}{(- 3 x - 8) \cdot 7} - \frac{1}{7} \cdot \frac{- 3 x - 8}{- 3 x - 8}}{\frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 1.3.2

Multiply $\frac{1}{7}$ by $\frac{- 3 x - 8}{- 3 x - 8}$ .

$lim_{x \to - 5} \frac{\frac{7}{(- 3 x - 8) \cdot 7} - \frac{- 3 x - 8}{7 (- 3 x - 8)}}{\frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 1.3.3

Reorder the factors of $(- 3 x - 8) \cdot 7$ .

$lim_{x \to - 5} \frac{\frac{7}{7 (- 3 x - 8)} - \frac{- 3 x - 8}{7 (- 3 x - 8)}}{\frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to - 5} \frac{\frac{7 - (- 3 x - 8)}{7 (- 3 x - 8)}}{\frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to - 5} \frac{7 - (- 3 x - 8)}{7 (- 3 x - 8)} \cdot \frac{1}{\frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 2.1.2

Multiply $\frac{7 - (- 3 x - 8)}{7 (- 3 x - 8)}$ by $\frac{1}{\frac{1}{- 3 x - 9} - \frac{1}{6}}$ .

$lim_{x \to - 5} \frac{7 - (- 3 x - 8)}{7 (- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

Step 2.2

Move the term $\frac{1}{7}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{7} lim_{x \to - 5} \frac{7 - (- 3 x - 8)}{(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

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.

$\frac{1}{7} \cdot \frac{lim_{x \to - 5} 7 - (- 3 x - 8)}{lim_{x \to - 5} (- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

Step 3.1.2

Evaluate the limits by plugging in $- 5$ for all occurrences of $x$ .

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Step 3.1.2.1

Evaluate the limit of $7 - (- 3 x - 8)$ by plugging in $- 5$ for $x$ .

$\frac{1}{7} \cdot \frac{7 - (- 3 \cdot - 5 - 8)}{lim_{x \to - 5} (- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

Step 3.1.2.2

Simplify each term.

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Step 3.1.2.2.1

Multiply $- 3$ by $- 5$ .

$\frac{1}{7} \cdot \frac{7 - (15 - 8)}{lim_{x \to - 5} (- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

Step 3.1.2.2.2

Subtract $8$ from $15$ .

$\frac{1}{7} \cdot \frac{7 - 1 \cdot 7}{lim_{x \to - 5} (- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

Step 3.1.2.2.3

Multiply $- 1$ by $7$ .

$\frac{1}{7} \cdot \frac{7 - 7}{lim_{x \to - 5} (- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

Step 3.1.2.3

Subtract $7$ from $7$ .

$\frac{1}{7} \cdot \frac{0}{lim_{x \to - 5} (- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

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 $- 5$ .

$\frac{1}{7} \cdot \frac{0}{lim_{x \to - 5} - 3 x - 8 \cdot lim_{x \to - 5} \frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 3.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot \frac{0}{(- lim_{x \to - 5} 3 x - lim_{x \to - 5} 8) \cdot lim_{x \to - 5} \frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 3.1.3.3

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

$\frac{1}{7} \cdot \frac{0}{(- 3 lim_{x \to - 5} x - lim_{x \to - 5} 8) \cdot lim_{x \to - 5} \frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 3.1.3.4

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

$\frac{1}{7} \cdot \frac{0}{(- 3 lim_{x \to - 5} x - 1 \cdot 8) \cdot lim_{x \to - 5} \frac{1}{- 3 x - 9} - \frac{1}{6}}$

Step 3.1.3.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot \frac{0}{(- 3 lim_{x \to - 5} x - 1 \cdot 8) \cdot (lim_{x \to - 5} \frac{1}{- 3 x - 9} - lim_{x \to - 5} \frac{1}{6})}$

Step 3.1.3.6

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot \frac{0}{(- 3 lim_{x \to - 5} x - 1 \cdot 8) \cdot (\frac{lim_{x \to - 5} 1}{lim_{x \to - 5} - 3 x - 9} - lim_{x \to - 5} \frac{1}{6})}$

Step 3.1.3.7

Evaluate the limit of $1$ which is constant as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot \frac{0}{(- 3 lim_{x \to - 5} x - 1 \cdot 8) \cdot (\frac{1}{lim_{x \to - 5} - 3 x - 9} - lim_{x \to - 5} \frac{1}{6})}$

Step 3.1.3.8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot \frac{0}{(- 3 lim_{x \to - 5} x - 1 \cdot 8) \cdot (\frac{1}{- lim_{x \to - 5} 3 x - lim_{x \to - 5} 9} - lim_{x \to - 5} \frac{1}{6})}$

Step 3.1.3.9

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

$\frac{1}{7} \cdot \frac{0}{(- 3 lim_{x \to - 5} x - 1 \cdot 8) \cdot (\frac{1}{- 3 lim_{x \to - 5} x - lim_{x \to - 5} 9} - lim_{x \to - 5} \frac{1}{6})}$

Step 3.1.3.10

Evaluate the limit of $9$ which is constant as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot \frac{0}{(- 3 lim_{x \to - 5} x - 1 \cdot 8) \cdot (\frac{1}{- 3 lim_{x \to - 5} x - 1 \cdot 9} - lim_{x \to - 5} \frac{1}{6})}$

Step 3.1.3.11

Evaluate the limit of $\frac{1}{6}$ which is constant as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot \frac{0}{(- 3 lim_{x \to - 5} x - 1 \cdot 8) \cdot (\frac{1}{- 3 lim_{x \to - 5} x - 1 \cdot 9} - \frac{1}{6})}$

Step 3.1.3.12

Evaluate the limits by plugging in $- 5$ for all occurrences of $x$ .

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Step 3.1.3.12.1

Evaluate the limit of $x$ by plugging in $- 5$ for $x$ .

$\frac{1}{7} \cdot \frac{0}{(- 3 \cdot - 5 - 1 \cdot 8) \cdot (\frac{1}{- 3 lim_{x \to - 5} x - 1 \cdot 9} - \frac{1}{6})}$

Step 3.1.3.12.2

Evaluate the limit of $x$ by plugging in $- 5$ for $x$ .

$\frac{1}{7} \cdot \frac{0}{(- 3 \cdot - 5 - 1 \cdot 8) \cdot (\frac{1}{- 3 \cdot - 5 - 1 \cdot 9} - \frac{1}{6})}$

Step 3.1.3.13

Simplify the answer.

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Step 3.1.3.13.1

Simplify each term.

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Step 3.1.3.13.1.1

Multiply $- 3$ by $- 5$ .

$\frac{1}{7} \cdot \frac{0}{(15 - 1 \cdot 8) \cdot (\frac{1}{- 3 \cdot - 5 - 1 \cdot 9} - \frac{1}{6})}$

Step 3.1.3.13.1.2

Multiply $- 1$ by $8$ .

$\frac{1}{7} \cdot \frac{0}{(15 - 8) \cdot (\frac{1}{- 3 \cdot - 5 - 1 \cdot 9} - \frac{1}{6})}$

Step 3.1.3.13.2

Subtract $8$ from $15$ .

$\frac{1}{7} \cdot \frac{0}{7 \cdot (\frac{1}{- 3 \cdot - 5 - 1 \cdot 9} - \frac{1}{6})}$

Step 3.1.3.13.3

Simplify the denominator.

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Step 3.1.3.13.3.1

Multiply $- 3$ by $- 5$ .

$\frac{1}{7} \cdot \frac{0}{7 \cdot (\frac{1}{15 - 1 \cdot 9} - \frac{1}{6})}$

Step 3.1.3.13.3.2

Multiply $- 1$ by $9$ .

$\frac{1}{7} \cdot \frac{0}{7 \cdot (\frac{1}{15 - 9} - \frac{1}{6})}$

Step 3.1.3.13.3.3

Subtract $9$ from $15$ .

$\frac{1}{7} \cdot \frac{0}{7 \cdot (\frac{1}{6} - \frac{1}{6})}$

Step 3.1.3.13.4

Combine the numerators over the common denominator.

$\frac{1}{7} \cdot \frac{0}{7 \cdot \frac{1 - 1}{6}}$

Step 3.1.3.13.5

Subtract $1$ from $1$ .

$\frac{1}{7} \cdot \frac{0}{7 \cdot \frac{0}{6}}$

Step 3.1.3.13.6

Divide $0$ by $6$ .

$\frac{1}{7} \cdot \frac{0}{7 \cdot 0}$

Step 3.1.3.13.7

Multiply $7$ by $0$ .

$\frac{1}{7} \cdot \frac{0}{0}$

Step 3.1.3.13.8

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

Undefined

$\frac{1}{7} \cdot \frac{0}{0}$

Step 3.1.3.14

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

Undefined

$\frac{1}{7} \cdot \frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{1}{7} \cdot \frac{0}{0}$

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.

$lim_{x \to - 5} \frac{7 - (- 3 x - 8)}{(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})} = lim_{x \to - 5} \frac{\frac{d}{d x} [7 - (- 3 x - 8)]}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$\frac{1}{7} lim_{x \to - 5} \frac{\frac{d}{d x} [7 - (- 3 x - 8)]}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.2

By the Sum Rule, the derivative of $7 - (- 3 x - 8)$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- (- 3 x - 8)]$ .

$\frac{1}{7} lim_{x \to - 5} \frac{\frac{d}{d x} [7] + \frac{d}{d x} [- (- 3 x - 8)]}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{1}{7} lim_{x \to - 5} \frac{0 + \frac{d}{d x} [- (- 3 x - 8)]}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.4

Evaluate $\frac{d}{d x} [- (- 3 x - 8)]$ .

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Step 3.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (- 3 x - 8)$ with respect to $x$ is $- \frac{d}{d x} [- 3 x - 8]$ .

$\frac{1}{7} lim_{x \to - 5} \frac{0 - \frac{d}{d x} [- 3 x - 8]}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.4.2

By the Sum Rule, the derivative of $- 3 x - 8$ with respect to $x$ is $\frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 8]$ .

$\frac{1}{7} lim_{x \to - 5} \frac{0 - (\frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 8])}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.4.3

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

$\frac{1}{7} lim_{x \to - 5} \frac{0 - (- 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 8])}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.4.4

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

$\frac{1}{7} lim_{x \to - 5} \frac{0 - (- 3 \cdot 1 + \frac{d}{d x} [- 8])}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.4.5

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

$\frac{1}{7} lim_{x \to - 5} \frac{0 - (- 3 \cdot 1 + 0)}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.4.6

Multiply $- 3$ by $1$ .

$\frac{1}{7} lim_{x \to - 5} \frac{0 - (- 3 + 0)}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.4.7

Add $- 3$ and $0$ .

$\frac{1}{7} lim_{x \to - 5} \frac{0 - - 3}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.4.8

Multiply $- 1$ by $- 3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{0 + 3}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

Step 3.3.5

Add $0$ and $3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{d}{d x} [(- 3 x - 8) (\frac{1}{- 3 x - 9} - \frac{1}{6})]}$

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 $f (x) = - 3 x - 8$ and $g (x) = \frac{1}{- 3 x - 9} - \frac{1}{6}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) \frac{d}{d x} [\frac{1}{- 3 x - 9} - \frac{1}{6}] + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.7

By the Sum Rule, the derivative of $\frac{1}{- 3 x - 9} - \frac{1}{6}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{- 3 x - 9}] + \frac{d}{d x} [- \frac{1}{6}]$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (\frac{d}{d x} [\frac{1}{- 3 x - 9}] + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.8

Rewrite $\frac{1}{- 3 x - 9}$ as ${(- 3 x - 9)}^{- 1}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (\frac{d}{d x} [{(- 3 x - 9)}^{- 1}] + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = - 3 x - 9$ .

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Step 3.3.9.1

To apply the Chain Rule, set $u$ as $- 3 x - 9$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [- 3 x - 9] + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.9.2

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (- u^{- 2} \frac{d}{d x} [- 3 x - 9] + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.9.3

Replace all occurrences of $u$ with $- 3 x - 9$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (- {(- 3 x - 9)}^{- 2} \frac{d}{d x} [- 3 x - 9] + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.10

By the Sum Rule, the derivative of $- 3 x - 9$ with respect to $x$ is $\frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 9]$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (- {(- 3 x - 9)}^{- 2} (\frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 9]) + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.11

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (- {(- 3 x - 9)}^{- 2} (- 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]) + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.12

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (- {(- 3 x - 9)}^{- 2} (- 3 \cdot 1 + \frac{d}{d x} [- 9]) + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.13

Multiply $- 3$ by $1$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (- {(- 3 x - 9)}^{- 2} (- 3 + \frac{d}{d x} [- 9]) + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.14

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (- {(- 3 x - 9)}^{- 2} (- 3 + 0) + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.15

Add $- 3$ and $0$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (- {(- 3 x - 9)}^{- 2} \cdot - 3 + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.16

Multiply $- 3$ by $- 1$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (3 {(- 3 x - 9)}^{- 2} + \frac{d}{d x} [- \frac{1}{6}]) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.17

Since $- \frac{1}{6}$ is constant with respect to $x$ , the derivative of $- \frac{1}{6}$ with respect to $x$ is $0$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) (3 {(- 3 x - 9)}^{- 2} + 0) + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.18

Add $3 {(- 3 x - 9)}^{- 2}$ and $0$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 3 x - 8) \cdot 3 {(- 3 x - 9)}^{- 2} + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.19

Move $3$ to the left of $- 3 x - 8$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{3 \cdot (- 3 x - 8) {(- 3 x - 9)}^{- 2} + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \frac{d}{d x} [- 3 x - 8]}$

Step 3.3.20

By the Sum Rule, the derivative of $- 3 x - 8$ with respect to $x$ is $\frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 8]$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{3 (- 3 x - 8) {(- 3 x - 9)}^{- 2} + (\frac{1}{- 3 x - 9} - \frac{1}{6}) (\frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 8])}$

Step 3.3.21

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{3 (- 3 x - 8) {(- 3 x - 9)}^{- 2} + (\frac{1}{- 3 x - 9} - \frac{1}{6}) (- 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 8])}$

Step 3.3.22

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{3 (- 3 x - 8) {(- 3 x - 9)}^{- 2} + (\frac{1}{- 3 x - 9} - \frac{1}{6}) (- 3 \cdot 1 + \frac{d}{d x} [- 8])}$

Step 3.3.23

Multiply $- 3$ by $1$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{3 (- 3 x - 8) {(- 3 x - 9)}^{- 2} + (\frac{1}{- 3 x - 9} - \frac{1}{6}) (- 3 + \frac{d}{d x} [- 8])}$

Step 3.3.24

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{3 (- 3 x - 8) {(- 3 x - 9)}^{- 2} + (\frac{1}{- 3 x - 9} - \frac{1}{6}) (- 3 + 0)}$

Step 3.3.25

Add $- 3$ and $0$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{3 (- 3 x - 8) {(- 3 x - 9)}^{- 2} + (\frac{1}{- 3 x - 9} - \frac{1}{6}) \cdot - 3}$

Step 3.3.26

Move $- 3$ to the left of $\frac{1}{- 3 x - 9} - \frac{1}{6}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{3 (- 3 x - 8) {(- 3 x - 9)}^{- 2} - 3 \cdot (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

Step 3.3.27

Simplify.

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Step 3.3.27.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{3 (- 3 x - 8) \frac{1}{{(- 3 x - 9)}^{2}} - 3 (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

Step 3.3.27.2

Apply the distributive property.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(3 \cdot - 3 x + 3 \cdot - 8) \frac{1}{{(- 3 x - 9)}^{2}} - 3 (\frac{1}{- 3 x - 9} - \frac{1}{6})}$

Step 3.3.27.3

Apply the distributive property.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(3 \cdot - 3 x + 3 \cdot - 8) \frac{1}{{(- 3 x - 9)}^{2}} - 3 \frac{1}{- 3 x - 9} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4

Combine terms.

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Step 3.3.27.4.1

Multiply $- 3$ by $3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x + 3 \cdot - 8) \frac{1}{{(- 3 x - 9)}^{2}} - 3 \frac{1}{- 3 x - 9} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.2

Multiply $3$ by $- 8$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - 3 \frac{1}{- 3 x - 9} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.3

Combine $- 3$ and $\frac{1}{- 3 x - 9}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} + \frac{- 3}{- 3 x - 9} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.4

Cancel the common factor of $- 3$ and $- 3 x - 9$ .

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Step 3.3.27.4.4.1

Factor $3$ out of $- 3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} + \frac{3 (- 1)}{- 3 x - 9} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.4.2

Cancel the common factors.

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Step 3.3.27.4.4.2.1

Factor $3$ out of $- 3 x$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} + \frac{3 (- 1)}{3 \cdot - 1 x - 9} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.4.2.2

Factor $3$ out of $- 9$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} + \frac{3 \cdot - 1}{3 \cdot - 1 x + 3 \cdot - 3} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.4.2.3

Factor $3$ out of $3 (- x) + 3 (- 3)$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} + \frac{3 (- 1)}{3 (- x - 3)} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.4.2.4

Cancel the common factor.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} + \frac{3 \cdot - 1}{3 (- x - 3)} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.4.2.5

Rewrite the expression.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} + \frac{- 1}{- x - 3} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.5

Move the negative in front of the fraction.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{1}{- x - 3} - 3 \cdot - 1 \frac{1}{6}}$

Step 3.3.27.4.6

Multiply $- 1$ by $- 3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{1}{- x - 3} + 3 (\frac{1}{6})}$

Step 3.3.27.4.7

Combine $3$ and $\frac{1}{6}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{1}{- x - 3} + \frac{3}{6}}$

Step 3.3.27.4.8

Cancel the common factor of $3$ and $6$ .

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Step 3.3.27.4.8.1

Factor $3$ out of $3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{1}{- x - 3} + \frac{3 (1)}{6}}$

Step 3.3.27.4.8.2

Cancel the common factors.

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Step 3.3.27.4.8.2.1

Factor $3$ out of $6$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{1}{- x - 3} + \frac{3 \cdot 1}{3 \cdot 2}}$

Step 3.3.27.4.8.2.2

Cancel the common factor.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{1}{- x - 3} + \frac{3 \cdot 1}{3 \cdot 2}}$

Step 3.3.27.4.8.2.3

Rewrite the expression.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{1}{- x - 3} + \frac{1}{2}}$

Step 3.3.27.4.9

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{1}{- x - 3} \cdot \frac{2}{2} + \frac{1}{2}}$

Step 3.3.27.4.10

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{1}{- x - 3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{- x - 3}{- x - 3}}$

Step 3.3.27.4.11

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

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Step 3.3.27.4.11.1

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{2}{(- x - 3) \cdot 2} + \frac{1}{2} \cdot \frac{- x - 3}{- x - 3}}$

Step 3.3.27.4.11.2

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{2}{(- x - 3) \cdot 2} + \frac{- x - 3}{2 (- x - 3)}}$

Step 3.3.27.4.11.3

Reorder the factors of $(- x - 3) \cdot 2$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} - \frac{2}{2 (- x - 3)} + \frac{- x - 3}{2 (- x - 3)}}$

Step 3.3.27.4.12

Combine the numerators over the common denominator.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} + \frac{- 2 - x - 3}{2 (- x - 3)}}$

Step 3.3.27.4.13

Subtract $3$ from $- 2$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} + \frac{- x - 5}{2 (- x - 3)}}$

Step 3.3.27.4.14

To write $(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{2 (- x - 3)}{2 (- x - 3)}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} \cdot \frac{2 (- x - 3)}{2 (- x - 3)} + \frac{- x - 5}{2 (- x - 3)}}$

Step 3.3.27.4.15

Combine $(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}}$ and $\frac{2 (- x - 3)}{2 (- x - 3)}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} \cdot 2 (- x - 3)}{2 (- x - 3)} + \frac{- x - 5}{2 (- x - 3)}}$

Step 3.3.27.4.16

Combine the numerators over the common denominator.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(- 9 x - 24) \frac{1}{{(- 3 x - 9)}^{2}} \cdot 2 (- x - 3) - x - 5}{2 (- x - 3)}}$

Step 3.3.27.4.17

Combine $2$ and $\frac{1}{{(- 3 x - 9)}^{2}}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(- 9 x - 24) \frac{2}{{(- 3 x - 9)}^{2}} (- x - 3) - x - 5}{2 (- x - 3)}}$

Step 3.3.27.5

Reorder terms.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(- 9 x - 24) (- x - 3) \frac{2}{{(- 3 x - 9)}^{2}} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.6

Simplify the denominator.

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Step 3.3.27.6.1

Factor $3$ out of $- 3 x - 9$ .

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Step 3.3.27.6.1.1

Factor $3$ out of $- 3 x$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(- 9 x - 24) (- x - 3) \frac{2}{{(3 \cdot - 1 x - 9)}^{2}} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.6.1.2

Factor $3$ out of $- 9$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(- 9 x - 24) (- x - 3) \frac{2}{{(3 \cdot - 1 x + 3 (- 3))}^{2}} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.6.1.3

Factor $3$ out of $3 (- x) + 3 (- 3)$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(- 9 x - 24) (- x - 3) \frac{2}{{(3 (- x - 3))}^{2}} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.6.2

Apply the product rule to $3 (- x - 3)$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(- 9 x - 24) (- x - 3) \frac{2}{3^{2} {(- x - 3)}^{2}} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.6.3

Raise $3$ to the power of $2$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(- 9 x - 24) (- x - 3) \frac{2}{9 {(- x - 3)}^{2}} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7

Simplify the numerator.

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Step 3.3.27.7.1

Cancel the common factor of $- (- x - 3)$ .

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Step 3.3.27.7.1.1

Factor $- (- x - 3)$ out of $(- 9 x - 24) (- x - 3)$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{- (- x - 3) (9 x + 24) \frac{2}{9 {(- x - 3)}^{2}} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.1.2

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{- (- x - 3) (9 x + 24) \frac{2}{- (- x - 3) \cdot - 9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.1.3

Cancel the common factor.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{- (- x - 3) (9 x + 24) \frac{2}{- (- x - 3) \cdot - 9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.1.4

Rewrite the expression.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(9 x + 24) \frac{2}{- 9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.2

Move the negative in front of the fraction.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{(9 x + 24) \cdot - 1 \frac{2}{9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.3

Apply the distributive property.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{9 x \cdot - 1 \frac{2}{9 (- x - 3)} + 24 \cdot - 1 \frac{2}{9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.4

Cancel the common factor of $9$ .

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Step 3.3.27.7.4.1

Move the leading negative in $- \frac{2}{9 (- x - 3)}$ into the numerator.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{9 x \frac{- 2}{9 (- x - 3)} + 24 \cdot - 1 \frac{2}{9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.4.2

Factor $9$ out of $9 x$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{9 (x) \frac{- 2}{9 (- x - 3)} + 24 \cdot - 1 \frac{2}{9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.4.3

Cancel the common factor.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{9 x \frac{- 2}{9 (- x - 3)} + 24 \cdot - 1 \frac{2}{9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.4.4

Rewrite the expression.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{x \frac{- 2}{- x - 3} + 24 \cdot - 1 \frac{2}{9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.5

Combine $x$ and $\frac{- 2}{- x - 3}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{x \cdot - 2}{- x - 3} + 24 \cdot - 1 \frac{2}{9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.6

Cancel the common factor of $3$ .

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Step 3.3.27.7.6.1

Move the leading negative in $- \frac{2}{9 (- x - 3)}$ into the numerator.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{x \cdot - 2}{- x - 3} + 24 \frac{- 2}{9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.6.2

Factor $3$ out of $24$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{x \cdot - 2}{- x - 3} + 3 (8) \frac{- 2}{9 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.6.3

Factor $3$ out of $9 (- x - 3)$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{x \cdot - 2}{- x - 3} + 3 (8) \frac{- 2}{3 \cdot 3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.6.4

Cancel the common factor.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{x \cdot - 2}{- x - 3} + 3 \cdot 8 \frac{- 2}{3 \cdot 3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.6.5

Rewrite the expression.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{x \cdot - 2}{- x - 3} + 8 \frac{- 2}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.7

Combine $8$ and $\frac{- 2}{3 (- x - 3)}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{x \cdot - 2}{- x - 3} + \frac{8 \cdot - 2}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.8

Multiply $8$ by $- 2$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{x \cdot - 2}{- x - 3} + \frac{- 16}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.9

Simplify each term.

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Step 3.3.27.7.9.1

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 2 \cdot x}{- x - 3} + \frac{- 16}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.9.2

Move the negative in front of the fraction.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{- \frac{2 \cdot x}{- x - 3} + \frac{- 16}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.9.3

Move the negative in front of the fraction.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{- \frac{2 x}{- x - 3} - \frac{16}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.10

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{- \frac{2 x}{- x - 3} \cdot \frac{3}{3} - \frac{16}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.11

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

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Step 3.3.27.7.11.1

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{- \frac{2 x \cdot 3}{(- x - 3) \cdot 3} - \frac{16}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.11.2

Reorder the factors of $(- x - 3) \cdot 3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{- \frac{2 x \cdot 3}{3 (- x - 3)} - \frac{16}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.12

Combine the numerators over the common denominator.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 2 x \cdot 3 - 16}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.13

Simplify the numerator.

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Step 3.3.27.7.13.1

Factor $2$ out of $- 2 x \cdot 3 - 16$ .

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Step 3.3.27.7.13.1.1

Factor $2$ out of $- 2 x \cdot 3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{2 \cdot - 1 x \cdot 3 - 16}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.13.1.2

Factor $2$ out of $- 16$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{2 \cdot - 1 x \cdot 3 + 2 (- 8)}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.13.1.3

Factor $2$ out of $2 (- x \cdot 3) + 2 (- 8)$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{2 (- x \cdot 3 - 8)}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.13.2

Multiply $3$ by $- 1$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{2 (- 3 x - 8)}{3 (- x - 3)} - x - 5}{2 (- x - 3)}}$

Step 3.3.27.7.14

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{2 (- 3 x - 8)}{3 (- x - 3)} - x \cdot \frac{3 (- x - 3)}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.15

Combine $- x$ and $\frac{3 (- x - 3)}{3 (- x - 3)}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{2 (- 3 x - 8)}{3 (- x - 3)} + \frac{- x \cdot 3 (- x - 3)}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.16

Combine the numerators over the common denominator.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{2 (- 3 x - 8) - x \cdot 3 (- x - 3)}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17

Simplify the numerator.

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Step 3.3.27.7.17.1

Apply the distributive property.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{2 \cdot - 3 x + 2 \cdot - 8 - x \cdot 3 (- x - 3)}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.2

Multiply $- 3$ by $2$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x + 2 \cdot - 8 - x \cdot 3 (- x - 3)}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.3

Multiply $2$ by $- 8$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 - x \cdot 3 (- x - 3)}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.4

Apply the distributive property.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 - x (3 \cdot - 1 x + 3 \cdot - 3)}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.5

Multiply $- 1$ by $3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 - x (- 3 x + 3 \cdot - 3)}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.6

Multiply $3$ by $- 3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 - x (- 3 x - 9)}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.7

Apply the distributive property.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 - x \cdot - 3 x - x \cdot - 9}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.8

Rewrite using the commutative property of multiplication.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 - 1 \cdot - 3 x \cdot x - x \cdot - 9}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.9

Multiply $- 9$ by $- 1$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 - 1 \cdot - 3 x \cdot x + 9 x}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.10

Simplify each term.

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Step 3.3.27.7.17.10.1

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

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Step 3.3.27.7.17.10.1.1

Move $x$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 - 1 \cdot - 3 x \cdot x + 9 x}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.10.1.2

Multiply $x$ by $x$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 - 1 \cdot - 3 x^{2} + 9 x}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.10.2

Multiply $- 1$ by $- 3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{- 6 x - 16 + 3 x^{2} + 9 x}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.11

Add $- 6 x$ and $9 x$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x - 16 + 3 x^{2}}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.17.12

Reorder terms.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 3 x - 16}{3 (- x - 3)} - 5}{2 (- x - 3)}}$

Step 3.3.27.7.18

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3 (- x - 3)}{3 (- x - 3)}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 3 x - 16}{3 (- x - 3)} - 5 \cdot \frac{3 (- x - 3)}{3 (- x - 3)}}{2 (- x - 3)}}$

Step 3.3.27.7.19

Combine $- 5$ and $\frac{3 (- x - 3)}{3 (- x - 3)}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 3 x - 16}{3 (- x - 3)} + \frac{- 5 \cdot 3 (- x - 3)}{3 (- x - 3)}}{2 (- x - 3)}}$

Step 3.3.27.7.20

Combine the numerators over the common denominator.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 3 x - 16 - 5 \cdot 3 (- x - 3)}{3 (- x - 3)}}{2 (- x - 3)}}$

Step 3.3.27.7.21

Simplify the numerator.

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Step 3.3.27.7.21.1

Multiply $- 5$ by $3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 3 x - 16 - 15 (- x - 3)}{3 (- x - 3)}}{2 (- x - 3)}}$

Step 3.3.27.7.21.2

Apply the distributive property.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 3 x - 16 - 15 \cdot - 1 x - 15 \cdot - 3}{3 (- x - 3)}}{2 (- x - 3)}}$

Step 3.3.27.7.21.3

Multiply $- 1$ by $- 15$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 3 x - 16 + 15 x - 15 \cdot - 3}{3 (- x - 3)}}{2 (- x - 3)}}$

Step 3.3.27.7.21.4

Multiply $- 15$ by $- 3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 3 x - 16 + 15 x + 45}{3 (- x - 3)}}{2 (- x - 3)}}$

Step 3.3.27.7.21.5

Add $3 x$ and $15 x$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 18 x - 16 + 45}{3 (- x - 3)}}{2 (- x - 3)}}$

Step 3.3.27.7.21.6

Add $- 16$ and $45$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{\frac{3 x^{2} + 18 x + 29}{3 (- x - 3)}}{2 (- x - 3)}}$

Step 3.3.27.8

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{3 x^{2} + 18 x + 29}{3 (- x - 3)} \cdot \frac{1}{2 (- x - 3)}}$

Step 3.3.27.9

Multiply $\frac{3 x^{2} + 18 x + 29}{3 (- x - 3)} \cdot \frac{1}{2 (- x - 3)}$ .

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Step 3.3.27.9.1

Multiply $\frac{3 x^{2} + 18 x + 29}{3 (- x - 3)}$ by $\frac{1}{2 (- x - 3)}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{3 x^{2} + 18 x + 29}{3 (- x - 3) \cdot 2 (- x - 3)}}$

Step 3.3.27.9.2

Multiply $2$ by $3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{3 x^{2} + 18 x + 29}{6 (- x - 3) (- x - 3)}}$

Step 3.3.27.9.3

Raise $- x - 3$ to the power of $1$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{3 x^{2} + 18 x + 29}{6 {(- x - 3)}^{1} (- x - 3)}}$

Step 3.3.27.9.4

Raise $- x - 3$ to the power of $1$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{3 x^{2} + 18 x + 29}{6 {(- x - 3)}^{1} {(- x - 3)}^{1}}}$

Step 3.3.27.9.5

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

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{3 x^{2} + 18 x + 29}{6 {(- x - 3)}^{1 + 1}}}$

Step 3.3.27.9.6

Add $1$ and $1$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3}{\frac{3 x^{2} + 18 x + 29}{6 {(- x - 3)}^{2}}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{7} lim_{x \to - 5} 3 \frac{6 {(- x - 3)}^{2}}{3 x^{2} + 18 x + 29}$

Step 3.5

Combine factors.

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Step 3.5.1

Combine $3$ and $\frac{6 {(- x - 3)}^{2}}{3 x^{2} + 18 x + 29}$ .

$\frac{1}{7} lim_{x \to - 5} \frac{3 \cdot 6 {(- x - 3)}^{2}}{3 x^{2} + 18 x + 29}$

Step 3.5.2

Multiply $6$ by $3$ .

$\frac{1}{7} lim_{x \to - 5} \frac{18 {(- x - 3)}^{2}}{3 x^{2} + 18 x + 29}$

Step 4

Evaluate the limit.

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Step 4.1

Move the term $18$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{7} \cdot 18 lim_{x \to - 5} \frac{{(- x - 3)}^{2}}{3 x^{2} + 18 x + 29}$

Step 4.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot 18 \frac{lim_{x \to - 5} {(- x - 3)}^{2}}{lim_{x \to - 5} 3 x^{2} + 18 x + 29}$

Step 4.3

Move the exponent $2$ from ${(- x - 3)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{1}{7} \cdot 18 \frac{{(lim_{x \to - 5} - x - 3)}^{2}}{lim_{x \to - 5} 3 x^{2} + 18 x + 29}$

Step 4.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot 18 \frac{{(- lim_{x \to - 5} x - lim_{x \to - 5} 3)}^{2}}{lim_{x \to - 5} 3 x^{2} + 18 x + 29}$

Step 4.5

Evaluate the limit of $3$ which is constant as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot 18 \frac{{(- lim_{x \to - 5} x - 1 \cdot 3)}^{2}}{lim_{x \to - 5} 3 x^{2} + 18 x + 29}$

Step 4.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot 18 \frac{{(- lim_{x \to - 5} x - 1 \cdot 3)}^{2}}{lim_{x \to - 5} 3 x^{2} + lim_{x \to - 5} 18 x + lim_{x \to - 5} 29}$

Step 4.7

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

$\frac{1}{7} \cdot 18 \frac{{(- lim_{x \to - 5} x - 1 \cdot 3)}^{2}}{3 lim_{x \to - 5} x^{2} + lim_{x \to - 5} 18 x + lim_{x \to - 5} 29}$

Step 4.8

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

$\frac{1}{7} \cdot 18 \frac{{(- lim_{x \to - 5} x - 1 \cdot 3)}^{2}}{3 {(lim_{x \to - 5} x)}^{2} + lim_{x \to - 5} 18 x + lim_{x \to - 5} 29}$

Step 4.9

Move the term $18$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{7} \cdot 18 \frac{{(- lim_{x \to - 5} x - 1 \cdot 3)}^{2}}{3 {(lim_{x \to - 5} x)}^{2} + 18 lim_{x \to - 5} x + lim_{x \to - 5} 29}$

Step 4.10

Evaluate the limit of $29$ which is constant as $x$ approaches $- 5$ .

$\frac{1}{7} \cdot 18 \frac{{(- lim_{x \to - 5} x - 1 \cdot 3)}^{2}}{3 {(lim_{x \to - 5} x)}^{2} + 18 lim_{x \to - 5} x + 29}$

Step 5

Evaluate the limits by plugging in $- 5$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $- 5$ for $x$ .

$\frac{1}{7} \cdot 18 \frac{{(5 - 1 \cdot 3)}^{2}}{3 {(lim_{x \to - 5} x)}^{2} + 18 lim_{x \to - 5} x + 29}$

Step 5.2

Evaluate the limit of $x$ by plugging in $- 5$ for $x$ .

$\frac{1}{7} \cdot 18 \frac{{(5 - 1 \cdot 3)}^{2}}{3 {(- 5)}^{2} + 18 lim_{x \to - 5} x + 29}$

Step 5.3

Evaluate the limit of $x$ by plugging in $- 5$ for $x$ .

$\frac{1}{7} \cdot 18 \frac{{(5 - 1 \cdot 3)}^{2}}{3 {(- 5)}^{2} + 18 \cdot - 5 + 29}$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{1}{7}$ and $18$ .

$\frac{18}{7} \cdot \frac{{(5 - 1 \cdot 3)}^{2}}{3 {(- 5)}^{2} + 18 \cdot - 5 + 29}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Multiply $- 1$ by $3$ .

$\frac{18}{7} \cdot \frac{{(5 - 3)}^{2}}{3 {(- 5)}^{2} + 18 \cdot - 5 + 29}$

Step 6.2.2

Subtract $3$ from $5$ .

$\frac{18}{7} \cdot \frac{2^{2}}{3 {(- 5)}^{2} + 18 \cdot - 5 + 29}$

Step 6.2.3

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

$\frac{18}{7} \cdot \frac{4}{3 {(- 5)}^{2} + 18 \cdot - 5 + 29}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Raise $- 5$ to the power of $2$ .

$\frac{18}{7} \cdot \frac{4}{3 \cdot 25 + 18 \cdot - 5 + 29}$

Step 6.3.2

Multiply $3$ by $25$ .

$\frac{18}{7} \cdot \frac{4}{75 + 18 \cdot - 5 + 29}$

Step 6.3.3

Multiply $18$ by $- 5$ .

$\frac{18}{7} \cdot \frac{4}{75 - 90 + 29}$

Step 6.3.4

Subtract $90$ from $75$ .

$\frac{18}{7} \cdot \frac{4}{- 15 + 29}$

Step 6.3.5

Add $- 15$ and $29$ .

$\frac{18}{7} \cdot \frac{4}{14}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Factor $2$ out of $18$ .

$\frac{2 (9)}{7} \cdot \frac{4}{14}$

Step 6.4.2

Factor $2$ out of $14$ .

$\frac{2 \cdot 9}{7} \cdot \frac{4}{2 \cdot 7}$

Step 6.4.3

Cancel the common factor.

$\frac{2 \cdot 9}{7} \cdot \frac{4}{2 \cdot 7}$

Step 6.4.4

Rewrite the expression.

$\frac{9}{7} \cdot \frac{4}{7}$

Step 6.5

Multiply $\frac{9}{7}$ by $\frac{4}{7}$ .

$\frac{9 \cdot 4}{7 \cdot 7}$

Step 6.6

Multiply $9$ by $4$ .

$\frac{36}{7 \cdot 7}$

Step 6.7

Multiply $7$ by $7$ .

$\frac{36}{49}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{36}{49}$

Decimal Form:

$0.73469387 \dots$