Calculus Examples

$lim_{x \to \frac{1}{2}} - 5 e^{- x^{4} - \frac{1}{4}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $- 5$ outside of the limit because it is constant with respect to $x$ .

$- 5 lim_{x \to \frac{1}{2}} e^{- x^{4} - \frac{1}{4}}$

Step 1.2

Move the limit into the exponent.

$- 5 e^{lim_{x \to \frac{1}{2}} - x^{4} - \frac{1}{4}}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$- 5 e^{- lim_{x \to \frac{1}{2}} x^{4} - lim_{x \to \frac{1}{2}} \frac{1}{4}}$

Step 1.4

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

$- 5 e^{- {(lim_{x \to \frac{1}{2}} x)}^{4} - lim_{x \to \frac{1}{2}} \frac{1}{4}}$

Step 1.5

Evaluate the limit of $\frac{1}{4}$ which is constant as $x$ approaches $\frac{1}{2}$ .

$- 5 e^{- {(lim_{x \to \frac{1}{2}} x)}^{4} - \frac{1}{4}}$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$- 5 e^{- {(\frac{1}{2})}^{4} - \frac{1}{4}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the product rule to $\frac{1}{2}$ .

$- 5 e^{- \frac{1^{4}}{2^{4}} - \frac{1}{4}}$

Step 3.1.2

One to any power is one.

$- 5 e^{- \frac{1}{2^{4}} - \frac{1}{4}}$

Step 3.1.3

Raise $2$ to the power of $4$ .

$- 5 e^{- \frac{1}{16} - \frac{1}{4}}$

Step 3.2

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 5 e^{- \frac{1}{16} - \frac{1}{4} \cdot \frac{4}{4}}$

Step 3.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$- 5 e^{- \frac{1}{16} - \frac{4}{4 \cdot 4}}$

Step 3.3.2

Multiply $4$ by $4$ .

$- 5 e^{- \frac{1}{16} - \frac{4}{16}}$

Step 3.4

Combine the numerators over the common denominator.

$- 5 e^{\frac{- 1 - 4}{16}}$

Step 3.5

Subtract $4$ from $- 1$ .

$- 5 e^{\frac{- 5}{16}}$

Step 3.6

Move the negative in front of the fraction.

$- 5 e^{- \frac{5}{16}}$

Step 3.7

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

$- 5 \frac{1}{e^{\frac{5}{16}}}$

Step 3.8

Combine $- 5$ and $\frac{1}{e^{\frac{5}{16}}}$ .

$\frac{- 5}{e^{\frac{5}{16}}}$

Step 3.9

Move the negative in front of the fraction.

$- \frac{5}{e^{\frac{5}{16}}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{e^{\frac{5}{16}}}$

Decimal Form:

$- 3.65807814 \dots$