Calculus Examples

$\frac{lim_{x \to - 8} 11 x^{3} - 11 x^{2} - 8 x}{9 - 11 x - 3 x^{3}}$

Step 1

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

$\frac{lim_{x \to - 8} 11 x^{3} - lim_{x \to - 8} 11 x^{2} - lim_{x \to - 8} 8 x}{9 - 11 x - 3 x^{3}}$

Step 2

Move the term $11$ outside of the limit because it is constant with respect to $x$ .

$\frac{11 lim_{x \to - 8} x^{3} - lim_{x \to - 8} 11 x^{2} - lim_{x \to - 8} 8 x}{9 - 11 x - 3 x^{3}}$

Step 3

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

$\frac{11 {(lim_{x \to - 8} x)}^{3} - lim_{x \to - 8} 11 x^{2} - lim_{x \to - 8} 8 x}{9 - 11 x - 3 x^{3}}$

Step 4

Move the term $11$ outside of the limit because it is constant with respect to $x$ .

$\frac{11 {(lim_{x \to - 8} x)}^{3} - 11 lim_{x \to - 8} x^{2} - lim_{x \to - 8} 8 x}{9 - 11 x - 3 x^{3}}$

Step 5

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

$\frac{11 {(lim_{x \to - 8} x)}^{3} - 11 {(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} 8 x}{9 - 11 x - 3 x^{3}}$

Step 6

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

$\frac{11 {(lim_{x \to - 8} x)}^{3} - 11 {(lim_{x \to - 8} x)}^{2} - 8 lim_{x \to - 8} x}{9 - 11 x - 3 x^{3}}$

Step 7

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

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Step 7.1

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

$\frac{11 {(- 8)}^{3} - 11 {(lim_{x \to - 8} x)}^{2} - 8 lim_{x \to - 8} x}{9 - 11 x - 3 x^{3}}$

Step 7.2

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

$\frac{11 {(- 8)}^{3} - 11 {(- 8)}^{2} - 8 lim_{x \to - 8} x}{9 - 11 x - 3 x^{3}}$

Step 7.3

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

$\frac{11 {(- 8)}^{3} - 11 {(- 8)}^{2} - 8 \cdot - 8}{9 - 11 x - 3 x^{3}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 8$ to the power of $3$ .

$\frac{11 \cdot - 512 - 11 {(- 8)}^{2} - 8 \cdot - 8}{9 - 11 x - 3 x^{3}}$

Step 8.1.2

Multiply $11$ by $- 512$ .

$\frac{- 5632 - 11 {(- 8)}^{2} - 8 \cdot - 8}{9 - 11 x - 3 x^{3}}$

Step 8.1.3

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

$\frac{- 5632 - 11 \cdot 64 - 8 \cdot - 8}{9 - 11 x - 3 x^{3}}$

Step 8.1.4

Multiply $- 11$ by $64$ .

$\frac{- 5632 - 704 - 8 \cdot - 8}{9 - 11 x - 3 x^{3}}$

Step 8.1.5

Multiply $- 8$ by $- 8$ .

$\frac{- 5632 - 704 + 64}{9 - 11 x - 3 x^{3}}$

Step 8.1.6

Subtract $704$ from $- 5632$ .

$\frac{- 6336 + 64}{9 - 11 x - 3 x^{3}}$

Step 8.1.7

Add $- 6336$ and $64$ .

$\frac{- 6272}{9 - 11 x - 3 x^{3}}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{6272}{9 - 11 x - 3 x^{3}}$