Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 4$ .

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

Step 2

Move the limit into the exponent.

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

Step 3

Move the exponent $3$ from ${(x - 4)}^{3}$ outside the limit using the Limits Power Rule.

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

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

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

Step 5

Evaluate the limit of $4$ which is constant as $x$ approaches $- 4$ .

$\frac{e^{lim_{x \to - 4} x}}{{(lim_{x \to - 4} x - 1 \cdot 4)}^{3}}$

Step 6

Evaluate the limits by plugging in $- 4$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$\frac{e^{- 4}}{{(lim_{x \to - 4} x - 1 \cdot 4)}^{3}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$\frac{e^{- 4}}{{(- 4 - 1 \cdot 4)}^{3}}$

Step 7

Simplify the answer.

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Step 7.1

Move $e^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(- 4 - 1 \cdot 4)}^{3} e^{4}}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Multiply $- 1$ by $4$ .

$\frac{1}{{(- 4 - 4)}^{3} e^{4}}$

Step 7.2.2

Subtract $4$ from $- 4$ .

$\frac{1}{{(- 8)}^{3} e^{4}}$

Step 7.2.3

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

$\frac{1}{- 512 e^{4}}$

Step 7.3

Move the negative in front of the fraction.

$- \frac{1}{512 e^{4}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{512 e^{4}}$

Decimal Form:

$- 3.57727322 \cdot 10^{- 5}$