Calculus Examples

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

Step 1

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

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

Step 2

Move the limit into the exponent.

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

Step 3

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

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

Step 4

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

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

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

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

Step 6

Move the limit under the radical sign.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Move $e^{- 2 \cdot 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.2

Combine $\frac{1}{\sqrt{8}}$ and $e^{2 \cdot 8}$ .

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

Step 8.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.4

Multiply $\frac{\sqrt{8}}{e^{2 \cdot 8}}$ by $1$ .

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

Step 8.5

Simplify the numerator.

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Step 8.5.1

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

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Step 8.5.1.1

Factor $4$ out of $8$ .

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

Step 8.5.1.2

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

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

Step 8.5.2

Pull terms out from under the radical.

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

Step 8.6

Multiply $2$ by $8$ .

$\frac{2 \sqrt{2}}{e^{16}}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{2}}{e^{16}}$

Decimal Form:

$3.1829754 \cdot 10^{- 7}$