Calculus Examples

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

Step 1

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

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

Step 2

Move the limit into the exponent.

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

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

$8 \cdot e^{- 8}$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

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

$8 \cdot \frac{1}{e^{8}}$

Step 5.2

Combine $8$ and $\frac{1}{e^{8}}$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.00268370 \dots$