Calculus Examples

$lim_{x \to - 8} \frac{e^{2} - x^{4}}{31 x - 2 x^{3}}$

Step 1

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

$\frac{lim_{x \to - 8} e^{2} - x^{4}}{lim_{x \to - 8} 31 x - 2 x^{3}}$

Step 2

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

$\frac{lim_{x \to - 8} e^{2} - lim_{x \to - 8} x^{4}}{lim_{x \to - 8} 31 x - 2 x^{3}}$

Step 3

Evaluate the limit of $e^{2}$ which is constant as $x$ approaches $- 8$ .

$\frac{e^{2} - lim_{x \to - 8} x^{4}}{lim_{x \to - 8} 31 x - 2 x^{3}}$

Step 4

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

$\frac{e^{2} - {(lim_{x \to - 8} x)}^{4}}{lim_{x \to - 8} 31 x - 2 x^{3}}$

Step 5

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

$\frac{e^{2} - {(lim_{x \to - 8} x)}^{4}}{lim_{x \to - 8} 31 x - lim_{x \to - 8} 2 x^{3}}$

Step 6

Move the term $31$ outside of the limit because it is constant with respect to $x$ .

$\frac{e^{2} - {(lim_{x \to - 8} x)}^{4}}{31 lim_{x \to - 8} x - lim_{x \to - 8} 2 x^{3}}$

Step 7

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

$\frac{e^{2} - {(lim_{x \to - 8} x)}^{4}}{31 lim_{x \to - 8} x - 2 lim_{x \to - 8} x^{3}}$

Step 8

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

$\frac{e^{2} - {(lim_{x \to - 8} x)}^{4}}{31 lim_{x \to - 8} x - 2 {(lim_{x \to - 8} x)}^{3}}$

Step 9

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

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Step 9.1

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

$\frac{e^{2} - {(- 8)}^{4}}{31 lim_{x \to - 8} x - 2 {(lim_{x \to - 8} x)}^{3}}$

Step 9.2

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

$\frac{e^{2} - {(- 8)}^{4}}{31 \cdot - 8 - 2 {(lim_{x \to - 8} x)}^{3}}$

Step 9.3

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

$\frac{e^{2} - {(- 8)}^{4}}{31 \cdot - 8 - 2 {(- 8)}^{3}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Rewrite ${(- 8)}^{4}$ as ${({(- 8)}^{2})}^{2}$ .

$\frac{e^{2} - {({(- 8)}^{2})}^{2}}{31 \cdot - 8 - 2 {(- 8)}^{3}}$

Step 10.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e$ and $b = {(- 8)}^{2}$ .

$\frac{(e + {(- 8)}^{2}) (e - {(- 8)}^{2})}{31 \cdot - 8 - 2 {(- 8)}^{3}}$

Step 10.1.3

Simplify.

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Step 10.1.3.1

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

$\frac{(e + 64) (e - {(- 8)}^{2})}{31 \cdot - 8 - 2 {(- 8)}^{3}}$

Step 10.1.3.2

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

$\frac{(e + 64) (e - 1 \cdot 64)}{31 \cdot - 8 - 2 {(- 8)}^{3}}$

Step 10.1.3.3

Multiply $- 1$ by $64$ .

$\frac{(e + 64) (e - 64)}{31 \cdot - 8 - 2 {(- 8)}^{3}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $31$ by $- 8$ .

$\frac{(e + 64) (e - 64)}{- 248 - 2 {(- 8)}^{3}}$

Step 10.2.2

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

$\frac{(e + 64) (e - 64)}{- 248 - 2 \cdot - 512}$

Step 10.2.3

Multiply $- 2$ by $- 512$ .

$\frac{(e + 64) (e - 64)}{- 248 + 1024}$

Step 10.2.4

Add $- 248$ and $1024$ .

$\frac{(e + 64) (e - 64)}{776}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{(e + 64) (e - 64)}{776}$

Decimal Form:

$- 5.26882853 \dots$