Calculus Examples

$lim_{x \to 8} (\sqrt{x}) e^{- \frac{x}{2}}$

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

Move the limit into the exponent.

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

Step 4

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

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

Step 5

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{lim_{x \to 8} x} \cdot e^{- \frac{1}{2} lim_{x \to 8} x}$

Step 6

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

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Step 6.1

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

$\sqrt{8} \cdot e^{- \frac{1}{2} lim_{x \to 8} x}$

Step 6.2

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

$\sqrt{8} \cdot e^{- \frac{1}{2} \cdot 8}$

Step 7

Simplify the answer.

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Step 7.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.1.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)} \cdot e^{- \frac{1}{2} \cdot 8}$

Step 7.1.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 2} \cdot e^{- \frac{1}{2} \cdot 8}$

Step 7.2

Pull terms out from under the radical.

$2 \sqrt{2} \cdot e^{- \frac{1}{2} \cdot 8}$

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$2 \sqrt{2} \cdot e^{\frac{- 1}{2} \cdot 8}$

Step 7.3.2

Factor $2$ out of $8$ .

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

Step 7.3.3

Cancel the common factor.

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

Step 7.3.4

Rewrite the expression.

$2 \sqrt{2} \cdot e^{- 1 \cdot 4}$

Step 7.4

Multiply $- 1$ by $4$ .

$2 \sqrt{2} \cdot e^{- 4}$

Step 7.5

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

$2 \sqrt{2} \cdot \frac{1}{e^{4}}$

Step 7.6

Multiply $2 \sqrt{2} \frac{1}{e^{4}}$ .

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Step 7.6.1

Combine $\frac{1}{e^{4}}$ and $2$ .

$\frac{2}{e^{4}} \sqrt{2}$

Step 7.6.2

Combine $\frac{2}{e^{4}}$ and $\sqrt{2}$ .

$\frac{2 \sqrt{2}}{e^{4}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{2}}{e^{4}}$

Decimal Form:

$0.05180444 \dots$