Calculus Examples

$lim_{x \to 8} x^{8} e^{- x^{7}}$

Step 1

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

$lim_{x \to 8} x^{8} \cdot lim_{x \to 8} e^{- x^{7}}$

Step 2

Move the exponent $8$ from $x^{8}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 8} x)}^{8} \cdot lim_{x \to 8} e^{- x^{7}}$

Step 3

Move the limit into the exponent.

${(lim_{x \to 8} x)}^{8} \cdot e^{lim_{x \to 8} - x^{7}}$

Step 4

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

${(lim_{x \to 8} x)}^{8} \cdot e^{- lim_{x \to 8} x^{7}}$

Step 5

Move the exponent $7$ from $x^{7}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 8} x)}^{8} \cdot e^{- {(lim_{x \to 8} x)}^{7}}$

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 $x$ .

$8^{8} \cdot e^{- {(lim_{x \to 8} x)}^{7}}$

Step 6.2

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

$8^{8} \cdot e^{- 8^{7}}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

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

$16777216 \cdot e^{- 8^{7}}$

Step 7.2

Raise $8$ to the power of $7$ .

$16777216 \cdot e^{- 1 \cdot 2097152}$

Step 7.3

Multiply $- 1$ by $2097152$ .

$16777216 \cdot e^{- 2097152}$

Step 7.4

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

$16777216 \cdot \frac{1}{e^{2097152}}$

Step 7.5

Combine $16777216$ and $\frac{1}{e^{2097152}}$ .

$\frac{16777216}{e^{2097152}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{16777216}{e^{2097152}}$

Decimal Form:

$0$