Calculus Examples

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

Step 1

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

$lim_{x \to 8} x^{3} \cdot lim_{x \to 8} e^{- x^{2}}$

Step 2

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

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

Step 3

Move the limit into the exponent.

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

Step 4

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

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

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

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

$8^{3} \cdot e^{- 8^{2}}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

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

$512 \cdot e^{- 8^{2}}$

Step 7.2

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

$512 \cdot e^{- 1 \cdot 64}$

Step 7.3

Multiply $- 1$ by $64$ .

$512 \cdot e^{- 64}$

Step 7.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$512 \cdot \frac{1}{e^{64}}$

Step 7.5

Combine $512$ and $\frac{1}{e^{64}}$ .

$\frac{512}{e^{64}}$