Calculus Examples

$lim_{x \to - 10} \frac{\frac{1}{- 9 x - 3} - \frac{1}{87}}{\frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to - 10} \frac{1}{- 9 x - 3} - \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{lim_{x \to - 10} \frac{1}{- 9 x - 3} - lim_{x \to - 10} \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.1.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 10$ .

$\frac{\frac{lim_{x \to - 10} 1}{lim_{x \to - 10} - 9 x - 3} - lim_{x \to - 10} \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $- 10$ .

$\frac{\frac{1}{lim_{x \to - 10} - 9 x - 3} - lim_{x \to - 10} \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{\frac{1}{- lim_{x \to - 10} 9 x - lim_{x \to - 10} 3} - lim_{x \to - 10} \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.1.5

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

$\frac{\frac{1}{- 9 lim_{x \to - 10} x - lim_{x \to - 10} 3} - lim_{x \to - 10} \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.1.6

Evaluate the limit of $3$ which is constant as $x$ approaches $- 10$ .

$\frac{\frac{1}{- 9 lim_{x \to - 10} x - 1 \cdot 3} - lim_{x \to - 10} \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.1.7

Evaluate the limit of $\frac{1}{87}$ which is constant as $x$ approaches $- 10$ .

$\frac{\frac{1}{- 9 lim_{x \to - 10} x - 1 \cdot 3} - \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$\frac{\frac{1}{- 9 \cdot - 10 - 1 \cdot 3} - \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify the denominator.

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Step 1.1.2.3.1.1

Multiply $- 9$ by $- 10$ .

$\frac{\frac{1}{90 - 1 \cdot 3} - \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $3$ .

$\frac{\frac{1}{90 - 3} - \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.3.1.3

Subtract $3$ from $90$ .

$\frac{\frac{1}{87} - \frac{1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.3.2

Combine the numerators over the common denominator.

$\frac{\frac{1 - 1}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.3.3

Subtract $1$ from $1$ .

$\frac{\frac{0}{87}}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.2.3.4

Divide $0$ by $87$ .

$\frac{0}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - \frac{1}{81}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{0}{lim_{x \to - 10} \frac{1}{- 9 x - 9} - lim_{x \to - 10} \frac{1}{81}}$

Step 1.1.3.1.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 10$ .

$\frac{0}{\frac{lim_{x \to - 10} 1}{lim_{x \to - 10} - 9 x - 9} - lim_{x \to - 10} \frac{1}{81}}$

Step 1.1.3.1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $- 10$ .

$\frac{0}{\frac{1}{lim_{x \to - 10} - 9 x - 9} - lim_{x \to - 10} \frac{1}{81}}$

Step 1.1.3.1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$\frac{0}{\frac{1}{- lim_{x \to - 10} 9 x - lim_{x \to - 10} 9} - lim_{x \to - 10} \frac{1}{81}}$

Step 1.1.3.1.5

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

$\frac{0}{\frac{1}{- 9 lim_{x \to - 10} x - lim_{x \to - 10} 9} - lim_{x \to - 10} \frac{1}{81}}$

Step 1.1.3.1.6

Evaluate the limit of $9$ which is constant as $x$ approaches $- 10$ .

$\frac{0}{\frac{1}{- 9 lim_{x \to - 10} x - 1 \cdot 9} - lim_{x \to - 10} \frac{1}{81}}$

Step 1.1.3.1.7

Evaluate the limit of $\frac{1}{81}$ which is constant as $x$ approaches $- 10$ .

$\frac{0}{\frac{1}{- 9 lim_{x \to - 10} x - 1 \cdot 9} - \frac{1}{81}}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$\frac{0}{\frac{1}{- 9 \cdot - 10 - 1 \cdot 9} - \frac{1}{81}}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify the denominator.

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Step 1.1.3.3.1.1

Multiply $- 9$ by $- 10$ .

$\frac{0}{\frac{1}{90 - 1 \cdot 9} - \frac{1}{81}}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $9$ .

$\frac{0}{\frac{1}{90 - 9} - \frac{1}{81}}$

Step 1.1.3.3.1.3

Subtract $9$ from $90$ .

$\frac{0}{\frac{1}{81} - \frac{1}{81}}$

Step 1.1.3.3.2

Combine the numerators over the common denominator.

$\frac{0}{\frac{1 - 1}{81}}$

Step 1.1.3.3.3

Subtract $1$ from $1$ .

$\frac{0}{\frac{0}{81}}$

Step 1.1.3.3.4

Divide $0$ by $81$ .

$\frac{0}{0}$

Step 1.1.3.3.5

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{x \to - 10} \frac{\frac{1}{- 9 x - 3} - \frac{1}{87}}{\frac{1}{- 9 x - 9} - \frac{1}{81}} = lim_{x \to - 10} \frac{\frac{d}{d x} [\frac{1}{- 9 x - 3} - \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 10} \frac{\frac{d}{d x} [\frac{1}{- 9 x - 3} - \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.2

By the Sum Rule, the derivative of $\frac{1}{- 9 x - 3} - \frac{1}{87}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{- 9 x - 3}] + \frac{d}{d x} [- \frac{1}{87}]$ .

$lim_{x \to - 10} \frac{\frac{d}{d x} [\frac{1}{- 9 x - 3}] + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [\frac{1}{- 9 x - 3}]$ .

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Step 1.3.3.1

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

$lim_{x \to - 10} \frac{\frac{d}{d x} [{(- 9 x - 3)}^{- 1}] + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.2

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 $g (x) = - 9 x - 3$ .

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Step 1.3.3.2.1

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

$lim_{x \to - 10} \frac{\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [- 9 x - 3] + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.2.2

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

$lim_{x \to - 10} \frac{- u^{- 2} \frac{d}{d x} [- 9 x - 3] + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.2.3

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

$lim_{x \to - 10} \frac{- {(- 9 x - 3)}^{- 2} \frac{d}{d x} [- 9 x - 3] + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.3

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

$lim_{x \to - 10} \frac{- {(- 9 x - 3)}^{- 2} (\frac{d}{d x} [- 9 x] + \frac{d}{d x} [- 3]) + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.4

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$lim_{x \to - 10} \frac{- {(- 9 x - 3)}^{- 2} (- 9 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.5

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

$lim_{x \to - 10} \frac{- {(- 9 x - 3)}^{- 2} (- 9 \cdot 1 + \frac{d}{d x} [- 3]) + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$lim_{x \to - 10} \frac{- {(- 9 x - 3)}^{- 2} (- 9 \cdot 1 + 0) + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.7

Multiply $- 9$ by $1$ .

$lim_{x \to - 10} \frac{- {(- 9 x - 3)}^{- 2} (- 9 + 0) + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.8

Add $- 9$ and $0$ .

$lim_{x \to - 10} \frac{- {(- 9 x - 3)}^{- 2} \cdot - 9 + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.3.9

Multiply $- 9$ by $- 1$ .

$lim_{x \to - 10} \frac{9 {(- 9 x - 3)}^{- 2} + \frac{d}{d x} [- \frac{1}{87}]}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.4

Since $- \frac{1}{87}$ is constant with respect to $x$ , the derivative of $- \frac{1}{87}$ with respect to $x$ is $0$ .

$lim_{x \to - 10} \frac{9 {(- 9 x - 3)}^{- 2} + 0}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5

Simplify.

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Step 1.3.5.1

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

$lim_{x \to - 10} \frac{9 \frac{1}{{(- 9 x - 3)}^{2}} + 0}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5.2

Combine terms.

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Step 1.3.5.2.1

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

$lim_{x \to - 10} \frac{\frac{9}{{(- 9 x - 3)}^{2}} + 0}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5.2.2

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

$lim_{x \to - 10} \frac{\frac{9}{{(- 9 x - 3)}^{2}}}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5.3

Simplify the denominator.

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Step 1.3.5.3.1

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

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Step 1.3.5.3.1.1

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

$lim_{x \to - 10} \frac{\frac{9}{{(3 (- 3 x) - 3)}^{2}}}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5.3.1.2

Factor $3$ out of $- 3$ .

$lim_{x \to - 10} \frac{\frac{9}{{(3 (- 3 x) + 3 (- 1))}^{2}}}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5.3.1.3

Factor $3$ out of $3 (- 3 x) + 3 (- 1)$ .

$lim_{x \to - 10} \frac{\frac{9}{{(3 (- 3 x - 1))}^{2}}}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5.3.2

Apply the product rule to $3 (- 3 x - 1)$ .

$lim_{x \to - 10} \frac{\frac{9}{3^{2} {(- 3 x - 1)}^{2}}}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5.3.3

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

$lim_{x \to - 10} \frac{\frac{9}{9 {(- 3 x - 1)}^{2}}}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5.4

Cancel the common factor of $9$ .

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Step 1.3.5.4.1

Cancel the common factor.

$lim_{x \to - 10} \frac{\frac{9}{9 {(- 3 x - 1)}^{2}}}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.5.4.2

Rewrite the expression.

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{d}{d x} [\frac{1}{- 9 x - 9} - \frac{1}{81}]}$

Step 1.3.6

By the Sum Rule, the derivative of $\frac{1}{- 9 x - 9} - \frac{1}{81}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{- 9 x - 9}] + \frac{d}{d x} [- \frac{1}{81}]$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{d}{d x} [\frac{1}{- 9 x - 9}] + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7

Evaluate $\frac{d}{d x} [\frac{1}{- 9 x - 9}]$ .

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Step 1.3.7.1

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{d}{d x} [{(- 9 x - 9)}^{- 1}] + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.2

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 $g (x) = - 9 x - 9$ .

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Step 1.3.7.2.1

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [- 9 x - 9] + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.2.2

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{- u^{- 2} \frac{d}{d x} [- 9 x - 9] + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.2.3

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{- {(- 9 x - 9)}^{- 2} \frac{d}{d x} [- 9 x - 9] + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.3

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{- {(- 9 x - 9)}^{- 2} (\frac{d}{d x} [- 9 x] + \frac{d}{d x} [- 9]) + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.4

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{- {(- 9 x - 9)}^{- 2} (- 9 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]) + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.5

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{- {(- 9 x - 9)}^{- 2} (- 9 \cdot 1 + \frac{d}{d x} [- 9]) + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.6

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{- {(- 9 x - 9)}^{- 2} (- 9 \cdot 1 + 0) + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.7

Multiply $- 9$ by $1$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{- {(- 9 x - 9)}^{- 2} (- 9 + 0) + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.8

Add $- 9$ and $0$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{- {(- 9 x - 9)}^{- 2} \cdot - 9 + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.7.9

Multiply $- 9$ by $- 1$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{9 {(- 9 x - 9)}^{- 2} + \frac{d}{d x} [- \frac{1}{81}]}$

Step 1.3.8

Since $- \frac{1}{81}$ is constant with respect to $x$ , the derivative of $- \frac{1}{81}$ with respect to $x$ is $0$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{9 {(- 9 x - 9)}^{- 2} + 0}$

Step 1.3.9

Simplify.

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Step 1.3.9.1

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{9 \frac{1}{{(- 9 x - 9)}^{2}} + 0}$

Step 1.3.9.2

Combine terms.

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Step 1.3.9.2.1

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9}{{(- 9 x - 9)}^{2}} + 0}$

Step 1.3.9.2.2

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9}{{(- 9 x - 9)}^{2}}}$

Step 1.3.9.3

Simplify the denominator.

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Step 1.3.9.3.1

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

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Step 1.3.9.3.1.1

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

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9}{{(9 (- x) - 9)}^{2}}}$

Step 1.3.9.3.1.2

Factor $9$ out of $- 9$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9}{{(9 (- x) + 9 (- 1))}^{2}}}$

Step 1.3.9.3.1.3

Factor $9$ out of $9 (- x) + 9 (- 1)$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9}{{(9 (- x - 1))}^{2}}}$

Step 1.3.9.3.2

Apply the product rule to $9 (- x - 1)$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9}{9^{2} {(- x - 1)}^{2}}}$

Step 1.3.9.3.3

Raise $9$ to the power of $2$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9}{81 {(- x - 1)}^{2}}}$

Step 1.3.9.4

Cancel the common factor of $9$ and $81$ .

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Step 1.3.9.4.1

Factor $9$ out of $9$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9 (1)}{81 {(- x - 1)}^{2}}}$

Step 1.3.9.4.2

Cancel the common factors.

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Step 1.3.9.4.2.1

Factor $9$ out of $81 {(- x - 1)}^{2}$ .

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9 (1)}{9 (9 {(- x - 1)}^{2})}}$

Step 1.3.9.4.2.2

Cancel the common factor.

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{9 \cdot 1}{9 (9 {(- x - 1)}^{2})}}$

Step 1.3.9.4.2.3

Rewrite the expression.

$lim_{x \to - 10} \frac{\frac{1}{{(- 3 x - 1)}^{2}}}{\frac{1}{9 {(- x - 1)}^{2}}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to - 10} \frac{1}{{(- 3 x - 1)}^{2}} (9 {(- x - 1)}^{2})$

Step 1.5

Combine factors.

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Step 1.5.1

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

$lim_{x \to - 10} \frac{9}{{(- 3 x - 1)}^{2}} {(- x - 1)}^{2}$

Step 1.5.2

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

$lim_{x \to - 10} \frac{9 {(- x - 1)}^{2}}{{(- 3 x - 1)}^{2}}$

Step 2

Evaluate the limit.

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Step 2.1

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

$9 lim_{x \to - 10} \frac{{(- x - 1)}^{2}}{{(- 3 x - 1)}^{2}}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 10$ .

$9 \frac{lim_{x \to - 10} {(- x - 1)}^{2}}{lim_{x \to - 10} {(- 3 x - 1)}^{2}}$

Step 2.3

Move the exponent $2$ from ${(- x - 1)}^{2}$ outside the limit using the Limits Power Rule.

$9 \frac{{(lim_{x \to - 10} - x - 1)}^{2}}{lim_{x \to - 10} {(- 3 x - 1)}^{2}}$

Step 2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$9 \frac{{(- lim_{x \to - 10} x - lim_{x \to - 10} 1)}^{2}}{lim_{x \to - 10} {(- 3 x - 1)}^{2}}$

Step 2.5

Evaluate the limit of $1$ which is constant as $x$ approaches $- 10$ .

$9 \frac{{(- lim_{x \to - 10} x - 1 \cdot 1)}^{2}}{lim_{x \to - 10} {(- 3 x - 1)}^{2}}$

Step 2.6

Move the exponent $2$ from ${(- 3 x - 1)}^{2}$ outside the limit using the Limits Power Rule.

$9 \frac{{(- lim_{x \to - 10} x - 1 \cdot 1)}^{2}}{{(lim_{x \to - 10} - 3 x - 1)}^{2}}$

Step 2.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 10$ .

$9 \frac{{(- lim_{x \to - 10} x - 1 \cdot 1)}^{2}}{{(- lim_{x \to - 10} 3 x - lim_{x \to - 10} 1)}^{2}}$

Step 2.8

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

$9 \frac{{(- lim_{x \to - 10} x - 1 \cdot 1)}^{2}}{{(- 3 lim_{x \to - 10} x - lim_{x \to - 10} 1)}^{2}}$

Step 2.9

Evaluate the limit of $1$ which is constant as $x$ approaches $- 10$ .

$9 \frac{{(- lim_{x \to - 10} x - 1 \cdot 1)}^{2}}{{(- 3 lim_{x \to - 10} x - 1 \cdot 1)}^{2}}$

Step 3

Evaluate the limits by plugging in $- 10$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$9 \frac{{(10 - 1 \cdot 1)}^{2}}{{(- 3 lim_{x \to - 10} x - 1 \cdot 1)}^{2}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $- 10$ for $x$ .

$9 \frac{{(10 - 1 \cdot 1)}^{2}}{{(- 3 \cdot - 10 - 1 \cdot 1)}^{2}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $1$ .

$9 \frac{{(10 - 1)}^{2}}{{(- 3 \cdot - 10 - 1 \cdot 1)}^{2}}$

Step 4.1.2

Subtract $1$ from $10$ .

$9 \frac{9^{2}}{{(- 3 \cdot - 10 - 1 \cdot 1)}^{2}}$

Step 4.1.3

Raise $9$ to the power of $2$ .

$9 \frac{81}{{(- 3 \cdot - 10 - 1 \cdot 1)}^{2}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $- 3$ by $- 10$ .

$9 \frac{81}{{(30 - 1 \cdot 1)}^{2}}$

Step 4.2.2

Multiply $- 1$ by $1$ .

$9 \frac{81}{{(30 - 1)}^{2}}$

Step 4.2.3

Subtract $1$ from $30$ .

$9 \frac{81}{29^{2}}$

Step 4.2.4

Raise $29$ to the power of $2$ .

$9 (\frac{81}{841})$

Step 4.3

Multiply $9 (\frac{81}{841})$ .

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Step 4.3.1

Combine $9$ and $\frac{81}{841}$ .

$\frac{9 \cdot 81}{841}$

Step 4.3.2

Multiply $9$ by $81$ .

$\frac{729}{841}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{729}{841}$

Decimal Form:

$0.86682520 \dots$