Calculus Examples

$lim_{x \to - 12} \frac{8 x^{3} - 16 x^{2} - 64 x}{x^{2} + 2 x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 12$ .

$\frac{lim_{x \to - 12} 8 x^{3} - 16 x^{2} - 64 x}{lim_{x \to - 12} x^{2} + 2 x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 12$ .

$\frac{lim_{x \to - 12} 8 x^{3} - lim_{x \to - 12} 16 x^{2} - lim_{x \to - 12} 64 x}{lim_{x \to - 12} x^{2} + 2 x}$

Step 3

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 lim_{x \to - 12} x^{3} - lim_{x \to - 12} 16 x^{2} - lim_{x \to - 12} 64 x}{lim_{x \to - 12} x^{2} + 2 x}$

Step 4

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

$\frac{8 {(lim_{x \to - 12} x)}^{3} - lim_{x \to - 12} 16 x^{2} - lim_{x \to - 12} 64 x}{lim_{x \to - 12} x^{2} + 2 x}$

Step 5

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 {(lim_{x \to - 12} x)}^{3} - 16 lim_{x \to - 12} x^{2} - lim_{x \to - 12} 64 x}{lim_{x \to - 12} x^{2} + 2 x}$

Step 6

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

$\frac{8 {(lim_{x \to - 12} x)}^{3} - 16 {(lim_{x \to - 12} x)}^{2} - lim_{x \to - 12} 64 x}{lim_{x \to - 12} x^{2} + 2 x}$

Step 7

Move the term $64$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 {(lim_{x \to - 12} x)}^{3} - 16 {(lim_{x \to - 12} x)}^{2} - 64 lim_{x \to - 12} x}{lim_{x \to - 12} x^{2} + 2 x}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 12$ .

$\frac{8 {(lim_{x \to - 12} x)}^{3} - 16 {(lim_{x \to - 12} x)}^{2} - 64 lim_{x \to - 12} x}{lim_{x \to - 12} x^{2} + lim_{x \to - 12} 2 x}$

Step 9

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

$\frac{8 {(lim_{x \to - 12} x)}^{3} - 16 {(lim_{x \to - 12} x)}^{2} - 64 lim_{x \to - 12} x}{{(lim_{x \to - 12} x)}^{2} + lim_{x \to - 12} 2 x}$

Step 10

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

$\frac{8 {(lim_{x \to - 12} x)}^{3} - 16 {(lim_{x \to - 12} x)}^{2} - 64 lim_{x \to - 12} x}{{(lim_{x \to - 12} x)}^{2} + 2 lim_{x \to - 12} x}$

Step 11

Evaluate the limits by plugging in $- 12$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $- 12$ for $x$ .

$\frac{8 {(- 12)}^{3} - 16 {(lim_{x \to - 12} x)}^{2} - 64 lim_{x \to - 12} x}{{(lim_{x \to - 12} x)}^{2} + 2 lim_{x \to - 12} x}$

Step 11.2

Evaluate the limit of $x$ by plugging in $- 12$ for $x$ .

$\frac{8 {(- 12)}^{3} - 16 {(- 12)}^{2} - 64 lim_{x \to - 12} x}{{(lim_{x \to - 12} x)}^{2} + 2 lim_{x \to - 12} x}$

Step 11.3

Evaluate the limit of $x$ by plugging in $- 12$ for $x$ .

$\frac{8 {(- 12)}^{3} - 16 {(- 12)}^{2} - 64 \cdot - 12}{{(lim_{x \to - 12} x)}^{2} + 2 lim_{x \to - 12} x}$

Step 11.4

Evaluate the limit of $x$ by plugging in $- 12$ for $x$ .

$\frac{8 {(- 12)}^{3} - 16 {(- 12)}^{2} - 64 \cdot - 12}{{(- 12)}^{2} + 2 lim_{x \to - 12} x}$

Step 11.5

Evaluate the limit of $x$ by plugging in $- 12$ for $x$ .

$\frac{8 {(- 12)}^{3} - 16 {(- 12)}^{2} - 64 \cdot - 12}{{(- 12)}^{2} + 2 \cdot - 12}$

$\frac{8 {(- 12)}^{3} - 16 {(- 12)}^{2} - 64 \cdot - 12}{{(- 12)}^{2} + 2 \cdot - 12}$

Step 12

Simplify the answer.

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Step 12.1

Cancel the common factor of $8 {(- 12)}^{3} - 16 {(- 12)}^{2} - 64 \cdot - 12$ and ${(- 12)}^{2} + 2 \cdot - 12$ .

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Step 12.1.1

Factor $- 12$ out of $8 {(- 12)}^{3}$ .

$\frac{- 12 (8 {(- 12)}^{2}) - 16 {(- 12)}^{2} - 64 \cdot - 12}{{(- 12)}^{2} + 2 \cdot - 12}$

Step 12.1.2

Factor $- 12$ out of $- 16 {(- 12)}^{2}$ .

$\frac{- 12 (8 {(- 12)}^{2}) - 12 (- 16 \cdot - 12) - 64 \cdot - 12}{{(- 12)}^{2} + 2 \cdot - 12}$

Step 12.1.3

Factor $- 12$ out of $- 12 (8 {(- 12)}^{2}) - 12 (- 16 \cdot - 12)$ .

$\frac{- 12 (8 {(- 12)}^{2} - 16 \cdot - 12) - 64 \cdot - 12}{{(- 12)}^{2} + 2 \cdot - 12}$

Step 12.1.4

Factor $- 12$ out of $- 64 \cdot - 12$ .

$\frac{- 12 (8 {(- 12)}^{2} - 16 \cdot - 12) - 12 \cdot - 64}{{(- 12)}^{2} + 2 \cdot - 12}$

Step 12.1.5

Factor $- 12$ out of $- 12 (8 {(- 12)}^{2} - 16 \cdot - 12) - 12 \cdot - 64$ .

$\frac{- 12 (8 {(- 12)}^{2} - 16 \cdot - 12 - 64)}{{(- 12)}^{2} + 2 \cdot - 12}$

Step 12.1.6

Cancel the common factors.

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Step 12.1.6.1

Factor $- 12$ out of ${(- 12)}^{2}$ .

$\frac{- 12 (8 {(- 12)}^{2} - 16 \cdot - 12 - 64)}{- 12 \cdot - 12 + 2 \cdot - 12}$

Step 12.1.6.2

Factor $- 12$ out of $2 \cdot - 12$ .

$\frac{- 12 (8 {(- 12)}^{2} - 16 \cdot - 12 - 64)}{- 12 \cdot - 12 - 12 \cdot 2}$

Step 12.1.6.3

Factor $- 12$ out of $- 12 \cdot - 12 - 12 \cdot 2$ .

$\frac{- 12 (8 {(- 12)}^{2} - 16 \cdot - 12 - 64)}{- 12 \cdot (- 12 + 2)}$

Step 12.1.6.4

Cancel the common factor.

$\frac{- 12 (8 {(- 12)}^{2} - 16 \cdot - 12 - 64)}{- 12 \cdot (- 12 + 2)}$

Step 12.1.6.5

Rewrite the expression.

$\frac{8 {(- 12)}^{2} - 16 \cdot - 12 - 64}{- 12 + 2}$

$\frac{8 {(- 12)}^{2} - 16 \cdot - 12 - 64}{- 12 + 2}$

$\frac{8 {(- 12)}^{2} - 16 \cdot - 12 - 64}{- 12 + 2}$

Step 12.2

Cancel the common factor of $8 {(- 12)}^{2} - 16 \cdot - 12 - 64$ and $- 12 + 2$ .

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Step 12.2.1

Factor $2$ out of $8 {(- 12)}^{2}$ .

$\frac{2 (4 {(- 12)}^{2}) - 16 \cdot - 12 - 64}{- 12 + 2}$

Step 12.2.2

Factor $2$ out of $- 16 \cdot - 12$ .

$\frac{2 (4 {(- 12)}^{2}) + 2 (- 8 \cdot - 12) - 64}{- 12 + 2}$

Step 12.2.3

Factor $2$ out of $2 (4 {(- 12)}^{2}) + 2 (- 8 \cdot - 12)$ .

$\frac{2 (4 {(- 12)}^{2} - 8 \cdot - 12) - 64}{- 12 + 2}$

Step 12.2.4

Factor $2$ out of $- 64$ .

$\frac{2 (4 {(- 12)}^{2} - 8 \cdot - 12) + 2 \cdot - 32}{- 12 + 2}$

Step 12.2.5

Factor $2$ out of $2 (4 {(- 12)}^{2} - 8 \cdot - 12) + 2 (- 32)$ .

$\frac{2 (4 {(- 12)}^{2} - 8 \cdot - 12 - 32)}{- 12 + 2}$

Step 12.2.6

Cancel the common factors.

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Step 12.2.6.1

Factor $2$ out of $- 12$ .

$\frac{2 (4 {(- 12)}^{2} - 8 \cdot - 12 - 32)}{2 \cdot - 6 + 2}$

Step 12.2.6.2

Factor $2$ out of $2$ .

$\frac{2 (4 {(- 12)}^{2} - 8 \cdot - 12 - 32)}{2 \cdot - 6 + 2 (1)}$

Step 12.2.6.3

Factor $2$ out of $2 \cdot - 6 + 2 (1)$ .

$\frac{2 (4 {(- 12)}^{2} - 8 \cdot - 12 - 32)}{2 \cdot (- 6 + 1)}$

Step 12.2.6.4

Cancel the common factor.

$\frac{2 (4 {(- 12)}^{2} - 8 \cdot - 12 - 32)}{2 \cdot (- 6 + 1)}$

Step 12.2.6.5

Rewrite the expression.

$\frac{4 {(- 12)}^{2} - 8 \cdot - 12 - 32}{- 6 + 1}$

$\frac{4 {(- 12)}^{2} - 8 \cdot - 12 - 32}{- 6 + 1}$

$\frac{4 {(- 12)}^{2} - 8 \cdot - 12 - 32}{- 6 + 1}$

Step 12.3

Simplify the numerator.

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Step 12.3.1

Raise $- 12$ to the power of $2$ .

$\frac{4 \cdot 144 - 8 \cdot - 12 - 32}{- 6 + 1}$

Step 12.3.2

Multiply $4$ by $144$ .

$\frac{576 - 8 \cdot - 12 - 32}{- 6 + 1}$

Step 12.3.3

Multiply $- 8$ by $- 12$ .

$\frac{576 + 96 - 32}{- 6 + 1}$

Step 12.3.4

Add $576$ and $96$ .

$\frac{672 - 32}{- 6 + 1}$

Step 12.3.5

Subtract $32$ from $672$ .

$\frac{640}{- 6 + 1}$

$\frac{640}{- 6 + 1}$

Step 12.4

Add $- 6$ and $1$ .

$\frac{640}{- 5}$

Step 12.5

Divide $640$ by $- 5$ .

$- 128$

$- 128$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$