Calculus Examples

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

Step 1

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

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

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

Step 3

Move the limit into the exponent.

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

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

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 $x$ .

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Multiply $2$ by $- 8$ .

$- 16 \cdot e^{2 \cdot - 8}$

Step 6.2

Multiply $2$ by $- 8$ .

$- 16 \cdot e^{- 16}$

Step 6.3

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

$- 16 \cdot \frac{1}{e^{16}}$

Step 6.4

Combine $- 16$ and $\frac{1}{e^{16}}$ .

$\frac{- 16}{e^{16}}$

Step 6.5

Move the negative in front of the fraction.

$- \frac{16}{e^{16}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{16}{e^{16}}$

Decimal Form:

$- 1.80056279 \cdot 10^{- 6}$