Calculus Examples

$\frac{lim_{x \to - 2} - x^{4} + 3 x^{3} + 9 x^{2} - 23 x + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{- lim_{x \to - 2} x^{4} + lim_{x \to - 2} 3 x^{3} + lim_{x \to - 2} 9 x^{2} - lim_{x \to - 2} 23 x + lim_{x \to - 2} 12}{x^{4} - 12 x^{2} - 16 x}$

Step 2

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{- {(lim_{x \to - 2} x)}^{4} + lim_{x \to - 2} 3 x^{3} + lim_{x \to - 2} 9 x^{2} - lim_{x \to - 2} 23 x + lim_{x \to - 2} 12}{x^{4} - 12 x^{2} - 16 x}$

Step 3

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

$\frac{- {(lim_{x \to - 2} x)}^{4} + 3 lim_{x \to - 2} x^{3} + lim_{x \to - 2} 9 x^{2} - lim_{x \to - 2} 23 x + lim_{x \to - 2} 12}{x^{4} - 12 x^{2} - 16 x}$

Step 4

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

$\frac{- {(lim_{x \to - 2} x)}^{4} + 3 {(lim_{x \to - 2} x)}^{3} + lim_{x \to - 2} 9 x^{2} - lim_{x \to - 2} 23 x + lim_{x \to - 2} 12}{x^{4} - 12 x^{2} - 16 x}$

Step 5

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

$\frac{- {(lim_{x \to - 2} x)}^{4} + 3 {(lim_{x \to - 2} x)}^{3} + 9 lim_{x \to - 2} x^{2} - lim_{x \to - 2} 23 x + lim_{x \to - 2} 12}{x^{4} - 12 x^{2} - 16 x}$

Step 6

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

$\frac{- {(lim_{x \to - 2} x)}^{4} + 3 {(lim_{x \to - 2} x)}^{3} + 9 {(lim_{x \to - 2} x)}^{2} - lim_{x \to - 2} 23 x + lim_{x \to - 2} 12}{x^{4} - 12 x^{2} - 16 x}$

Step 7

Move the term $23$ outside of the limit because it is constant with respect to $x$ .

$\frac{- {(lim_{x \to - 2} x)}^{4} + 3 {(lim_{x \to - 2} x)}^{3} + 9 {(lim_{x \to - 2} x)}^{2} - 23 lim_{x \to - 2} x + lim_{x \to - 2} 12}{x^{4} - 12 x^{2} - 16 x}$

Step 8

Evaluate the limit of $12$ which is constant as $x$ approaches $- 2$ .

$\frac{- {(lim_{x \to - 2} x)}^{4} + 3 {(lim_{x \to - 2} x)}^{3} + 9 {(lim_{x \to - 2} x)}^{2} - 23 lim_{x \to - 2} x + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 9

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{- {(- 2)}^{4} + 3 {(lim_{x \to - 2} x)}^{3} + 9 {(lim_{x \to - 2} x)}^{2} - 23 lim_{x \to - 2} x + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 9.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{- {(- 2)}^{4} + 3 {(- 2)}^{3} + 9 {(lim_{x \to - 2} x)}^{2} - 23 lim_{x \to - 2} x + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 9.3

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{- {(- 2)}^{4} + 3 {(- 2)}^{3} + 9 {(- 2)}^{2} - 23 lim_{x \to - 2} x + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 9.4

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{- {(- 2)}^{4} + 3 {(- 2)}^{3} + 9 {(- 2)}^{2} - 23 \cdot - 2 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $- 2$ to the power of $4$ .

$\frac{- 1 \cdot 16 + 3 {(- 2)}^{3} + 9 {(- 2)}^{2} - 23 \cdot - 2 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.2

Multiply $- 1$ by $16$ .

$\frac{- 16 + 3 {(- 2)}^{3} + 9 {(- 2)}^{2} - 23 \cdot - 2 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.3

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

$\frac{- 16 + 3 \cdot - 8 + 9 {(- 2)}^{2} - 23 \cdot - 2 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.4

Multiply $3$ by $- 8$ .

$\frac{- 16 - 24 + 9 {(- 2)}^{2} - 23 \cdot - 2 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.5

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

$\frac{- 16 - 24 + 9 \cdot 4 - 23 \cdot - 2 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.6

Multiply $9$ by $4$ .

$\frac{- 16 - 24 + 36 - 23 \cdot - 2 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.7

Multiply $- 23$ by $- 2$ .

$\frac{- 16 - 24 + 36 + 46 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.8

Subtract $24$ from $- 16$ .

$\frac{- 40 + 36 + 46 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.9

Add $- 40$ and $36$ .

$\frac{- 4 + 46 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.10

Add $- 4$ and $46$ .

$\frac{42 + 12}{x^{4} - 12 x^{2} - 16 x}$

Step 10.1.11

Add $42$ and $12$ .

$\frac{54}{x^{4} - 12 x^{2} - 16 x}$

Step 10.2

Factor $x$ out of $x^{4} - 12 x^{2} - 16 x$ .

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Step 10.2.1

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

$\frac{54}{x \cdot x^{3} - 12 x^{2} - 16 x}$

Step 10.2.2

Factor $x$ out of $- 12 x^{2}$ .

$\frac{54}{x \cdot x^{3} + x (- 12 x) - 16 x}$

Step 10.2.3

Factor $x$ out of $- 16 x$ .

$\frac{54}{x \cdot x^{3} + x (- 12 x) + x \cdot - 16}$

Step 10.2.4

Factor $x$ out of $x \cdot x^{3} + x (- 12 x)$ .

$\frac{54}{x (x^{3} - 12 x) + x \cdot - 16}$

Step 10.2.5

Factor $x$ out of $x (x^{3} - 12 x) + x \cdot - 16$ .

$\frac{54}{x (x^{3} - 12 x - 16)}$