Calculus Examples

$lim_{x \to 8} \frac{x^{3}}{e^{3 x}}$

Step 1

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

$\frac{lim_{x \to 8} x^{3}}{lim_{x \to 8} e^{3 x}}$

Step 2

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

$\frac{{(lim_{x \to 8} x)}^{3}}{lim_{x \to 8} e^{3 x}}$

Step 3

Move the limit into the exponent.

$\frac{{(lim_{x \to 8} x)}^{3}}{e^{lim_{x \to 8} 3 x}}$

Step 4

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

$\frac{{(lim_{x \to 8} x)}^{3}}{e^{3 lim_{x \to 8} x}}$

Step 5

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

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Step 5.1

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

$\frac{8^{3}}{e^{3 lim_{x \to 8} x}}$

Step 5.2

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

$\frac{8^{3}}{e^{3 \cdot 8}}$

Step 6

Simplify the answer.

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Step 6.1

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

$\frac{512}{e^{3 \cdot 8}}$

Step 6.2

Multiply $3$ by $8$ .

$\frac{512}{e^{24}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{512}{e^{24}}$

Decimal Form:

$1.93286888 \cdot 10^{- 8}$