Calculus Examples

$lim_{x \to - 8} \frac{x^{2} + 12 x + 32}{- 72 - 1 x + x^{2}}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to - 8} x^{2} + 12 x + 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

$\frac{lim_{x \to - 8} x^{2} + lim_{x \to - 8} 12 x + lim_{x \to - 8} 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2.2

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

$\frac{{(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} 12 x + lim_{x \to - 8} 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2.3

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

$\frac{{(lim_{x \to - 8} x)}^{2} + 12 lim_{x \to - 8} x + lim_{x \to - 8} 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2.4

Evaluate the limit of $32$ which is constant as $x$ approaches $- 8$ .

$\frac{{(lim_{x \to - 8} x)}^{2} + 12 lim_{x \to - 8} x + 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2.5

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 1.2.5.1

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

$\frac{{(- 8)}^{2} + 12 lim_{x \to - 8} x + 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2.5.2

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

$\frac{{(- 8)}^{2} + 12 \cdot - 8 + 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2.6

Simplify the answer.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

Raise $- 8$ to the power of $2$ .

$\frac{64 + 12 \cdot - 8 + 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2.6.1.2

Multiply $12$ by $- 8$ .

$\frac{64 - 96 + 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2.6.2

Subtract $96$ from $64$ .

$\frac{- 32 + 32}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.2.6.3

Add $- 32$ and $32$ .

$\frac{0}{lim_{x \to - 8} - 72 - 1 x + x^{2}}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

$\frac{0}{- lim_{x \to - 8} 72 - lim_{x \to - 8} x + lim_{x \to - 8} x^{2}}$

Step 1.3.2

Evaluate the limit of $72$ which is constant as $x$ approaches $- 8$ .

$\frac{0}{- 1 \cdot 72 - lim_{x \to - 8} x + lim_{x \to - 8} x^{2}}$

Step 1.3.3

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

$\frac{0}{- 1 \cdot 72 - lim_{x \to - 8} x + {(lim_{x \to - 8} x)}^{2}}$

Step 1.3.4

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 1.3.4.1

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

$\frac{0}{- 1 \cdot 72 + 8 + {(lim_{x \to - 8} x)}^{2}}$

Step 1.3.4.2

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

$\frac{0}{- 1 \cdot 72 + 8 + {(- 8)}^{2}}$

Step 1.3.5

Simplify the answer.

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Step 1.3.5.1

Simplify each term.

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Step 1.3.5.1.1

Multiply $- 1$ by $72$ .

$\frac{0}{- 72 + 8 + {(- 8)}^{2}}$

Step 1.3.5.1.2

Raise $- 8$ to the power of $2$ .

$\frac{0}{- 72 + 8 + 64}$

Step 1.3.5.2

Add $- 72$ and $8$ .

$\frac{0}{- 64 + 64}$

Step 1.3.5.3

Add $- 64$ and $64$ .

$\frac{0}{0}$

Step 1.3.5.4

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

Undefined

$\frac{0}{0}$

Step 1.3.6

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to - 8} \frac{x^{2} + 12 x + 32}{- 72 - 1 x + x^{2}} = lim_{x \to - 8} \frac{\frac{d}{d x} [x^{2} + 12 x + 32]}{\frac{d}{d x} [- 72 - 1 x + x^{2}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to - 8} \frac{\frac{d}{d x} [x^{2} + 12 x + 32]}{\frac{d}{d x} [- 72 - 1 x + x^{2}]}$

Step 3.2

By the Sum Rule, the derivative of $x^{2} + 12 x + 32$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [32]$ .

$lim_{x \to - 8} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [32]}{\frac{d}{d x} [- 72 - 1 x + x^{2}]}$

Step 3.3

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

$lim_{x \to - 8} \frac{2 x + \frac{d}{d x} [12 x] + \frac{d}{d x} [32]}{\frac{d}{d x} [- 72 - 1 x + x^{2}]}$

Step 3.4

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 3.4.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$lim_{x \to - 8} \frac{2 x + 12 \frac{d}{d x} [x] + \frac{d}{d x} [32]}{\frac{d}{d x} [- 72 - 1 x + x^{2}]}$

Step 3.4.2

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

$lim_{x \to - 8} \frac{2 x + 12 \cdot 1 + \frac{d}{d x} [32]}{\frac{d}{d x} [- 72 - 1 x + x^{2}]}$

Step 3.4.3

Multiply $12$ by $1$ .

$lim_{x \to - 8} \frac{2 x + 12 + \frac{d}{d x} [32]}{\frac{d}{d x} [- 72 - 1 x + x^{2}]}$

Step 3.5

Since $32$ is constant with respect to $x$ , the derivative of $32$ with respect to $x$ is $0$ .

$lim_{x \to - 8} \frac{2 x + 12 + 0}{\frac{d}{d x} [- 72 - 1 x + x^{2}]}$

Step 3.6

Add $2 x + 12$ and $0$ .

$lim_{x \to - 8} \frac{2 x + 12}{\frac{d}{d x} [- 72 - 1 x + x^{2}]}$

Step 3.7

By the Sum Rule, the derivative of $- 72 - 1 x + x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 72] + \frac{d}{d x} [- 1 x] + \frac{d}{d x} [x^{2}]$ .

$lim_{x \to - 8} \frac{2 x + 12}{\frac{d}{d x} [- 72] + \frac{d}{d x} [- 1 x] + \frac{d}{d x} [x^{2}]}$

Step 3.8

Since $- 72$ is constant with respect to $x$ , the derivative of $- 72$ with respect to $x$ is $0$ .

$lim_{x \to - 8} \frac{2 x + 12}{0 + \frac{d}{d x} [- 1 x] + \frac{d}{d x} [x^{2}]}$

Step 3.9

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

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Step 3.9.1

Rewrite $- 1 x$ as $- x$ .

$lim_{x \to - 8} \frac{2 x + 12}{0 + \frac{d}{d x} [- x] + \frac{d}{d x} [x^{2}]}$

Step 3.9.2

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

$lim_{x \to - 8} \frac{2 x + 12}{0 - \frac{d}{d x} [x] + \frac{d}{d x} [x^{2}]}$

Step 3.9.3

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

$lim_{x \to - 8} \frac{2 x + 12}{0 - 1 \cdot 1 + \frac{d}{d x} [x^{2}]}$

Step 3.9.4

Multiply $- 1$ by $1$ .

$lim_{x \to - 8} \frac{2 x + 12}{0 - 1 + \frac{d}{d x} [x^{2}]}$

Step 3.10

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

$lim_{x \to - 8} \frac{2 x + 12}{0 - 1 + 2 x}$

Step 3.11

Simplify.

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Step 3.11.1

Subtract $1$ from $0$ .

$lim_{x \to - 8} \frac{2 x + 12}{- 1 + 2 x}$

Step 3.11.2

Reorder terms.

$lim_{x \to - 8} \frac{2 x + 12}{2 x - 1}$

Step 4

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

$\frac{lim_{x \to - 8} 2 x + 12}{lim_{x \to - 8} 2 x - 1}$

Step 5

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

$\frac{lim_{x \to - 8} 2 x + lim_{x \to - 8} 12}{lim_{x \to - 8} 2 x - 1}$

Step 6

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

$\frac{2 lim_{x \to - 8} x + lim_{x \to - 8} 12}{lim_{x \to - 8} 2 x - 1}$

Step 7

Evaluate the limit of $12$ which is constant as $x$ approaches $- 8$ .

$\frac{2 lim_{x \to - 8} x + 12}{lim_{x \to - 8} 2 x - 1}$

Step 8

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

$\frac{2 lim_{x \to - 8} x + 12}{lim_{x \to - 8} 2 x - lim_{x \to - 8} 1}$

Step 9

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

$\frac{2 lim_{x \to - 8} x + 12}{2 lim_{x \to - 8} x - lim_{x \to - 8} 1}$

Step 10

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

$\frac{2 lim_{x \to - 8} x + 12}{2 lim_{x \to - 8} x - 1 \cdot 1}$

Step 11

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 11.1

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

$\frac{2 \cdot - 8 + 12}{2 lim_{x \to - 8} x - 1 \cdot 1}$

Step 11.2

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

$\frac{2 \cdot - 8 + 12}{2 \cdot - 8 - 1 \cdot 1}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Multiply $2$ by $- 8$ .

$\frac{- 16 + 12}{2 \cdot - 8 - 1 \cdot 1}$

Step 12.1.2

Add $- 16$ and $12$ .

$\frac{- 4}{2 \cdot - 8 - 1 \cdot 1}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Multiply $2$ by $- 8$ .

$\frac{- 4}{- 16 - 1 \cdot 1}$

Step 12.2.2

Multiply $- 1$ by $1$ .

$\frac{- 4}{- 16 - 1}$

Step 12.2.3

Subtract $1$ from $- 16$ .

$\frac{- 4}{- 17}$

Step 12.3

Dividing two negative values results in a positive value.

$\frac{4}{17}$