Calculus Examples

$lim_{x \to 0} - 2 e^{- 2 x^{4} - \frac{1}{10}}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

$- 2 lim_{x \to 0} e^{- 2 x^{4} - \frac{1}{10}}$

Step 1.2

Move the limit into the exponent.

$- 2 e^{lim_{x \to 0} - 2 x^{4} - \frac{1}{10}}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$- 2 e^{- lim_{x \to 0} 2 x^{4} - lim_{x \to 0} \frac{1}{10}}$

Step 1.4

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

$- 2 e^{- 2 lim_{x \to 0} x^{4} - lim_{x \to 0} \frac{1}{10}}$

Step 1.5

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$- 2 e^{- 2 {(lim_{x \to 0} x)}^{4} - lim_{x \to 0} \frac{1}{10}}$

Step 1.6

Evaluate the limit of $\frac{1}{10}$ which is constant as $x$ approaches $0$ .

$- 2 e^{- 2 {(lim_{x \to 0} x)}^{4} - \frac{1}{10}}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$- 2 e^{- 2 \cdot 0^{4} - \frac{1}{10}}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Raising $0$ to any positive power yields $0$ .

$- 2 e^{- 2 \cdot 0 - \frac{1}{10}}$

Step 3.1.2

Multiply $- 2$ by $0$ .

$- 2 e^{0 - \frac{1}{10}}$

Step 3.2

Subtract $\frac{1}{10}$ from $0$ .

$- 2 e^{- \frac{1}{10}}$

Step 3.3

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

$- 2 \frac{1}{e^{\frac{1}{10}}}$

Step 3.4

Combine $- 2$ and $\frac{1}{e^{\frac{1}{10}}}$ .

$\frac{- 2}{e^{\frac{1}{10}}}$

Step 3.5

Move the negative in front of the fraction.

$- \frac{2}{e^{\frac{1}{10}}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{e^{\frac{1}{10}}}$

Decimal Form:

$- 1.80967483 \dots$