Calculus Examples

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

Step 1

Rewrite $\sqrt{x} e^{- \frac{x}{2}}$ as $\frac{\sqrt{x}}{e^{\frac{x}{2}}}$ .

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}}}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to \infty} \sqrt{x}}{lim_{x \to \infty} e^{\frac{x}{2}}}$

Step 2.1.2

As $x$ approaches $\infty$ for radicals, the value goes to $\infty$ .

$\frac{\infty}{lim_{x \to \infty} e^{\frac{x}{2}}}$

Step 2.1.3

Since the exponent $\frac{x}{2}$ approaches $\infty$ , the quantity $e^{\frac{x}{2}}$ approaches $\infty$ .

$\frac{\infty}{\infty}$

Step 2.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{\infty}$

Step 2.2

Since $\frac{\infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\sqrt{x}]}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [\sqrt{x}]}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{\frac{1}{2}}]}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1}{2} - 1}}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.5

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.6

Combine the numerators over the common denominator.

$lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.7

Simplify the numerator.

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Step 2.3.7.1

Multiply $- 1$ by $2$ .

$lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{1 - 2}{2}}}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.7.2

Subtract $2$ from $1$ .

$lim_{x \to \infty} \frac{\frac{1}{2} x^{\frac{- 1}{2}}}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.8

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\frac{1}{2} x^{- \frac{1}{2}}}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.9

Simplify.

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Step 2.3.9.1

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

$lim_{x \to \infty} \frac{\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}}}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.9.2

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

$lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [e^{\frac{x}{2}}]}$

Step 2.3.10

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = \frac{x}{2}$ .

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Step 2.3.10.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x}{2}]}$

Step 2.3.10.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{e^{u} \frac{d}{d x} [\frac{x}{2}]}$

Step 2.3.10.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{e^{\frac{x}{2}} \frac{d}{d x} [\frac{x}{2}]}$

Step 2.3.11

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{e^{\frac{x}{2}} (\frac{1}{2} \frac{d}{d x} [x])}$

Step 2.3.12

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

$lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{e^{\frac{x}{2}}}{2} \frac{d}{d x} [x]}$

Step 2.3.13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{e^{\frac{x}{2}}}{2} \cdot 1}$

Step 2.3.14

Multiply $\frac{e^{\frac{x}{2}}}{2}$ by $1$ .

$lim_{x \to \infty} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{e^{\frac{x}{2}}}{2}}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \infty} \frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{2}{e^{\frac{x}{2}}}$

Step 2.5

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

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

Step 2.6

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

$lim_{x \to \infty} \frac{2}{2 \sqrt{x} e^{\frac{x}{2}}}$

Step 2.7

Cancel the common factor of $2$ .

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Step 2.7.1

Cancel the common factor.

$lim_{x \to \infty} \frac{2}{2 \sqrt{x} e^{\frac{x}{2}}}$

Step 2.7.2

Rewrite the expression.

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

Step 2.8

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

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

Step 2.9

Combine and simplify the denominator.

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Step 2.9.1

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

$lim_{x \to \infty} \frac{\sqrt{x}}{\sqrt{x} e^{\frac{x}{2}} \sqrt{x}}$

Step 2.9.2

Move $\sqrt{x}$ .

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

Step 2.9.3

Raise $\sqrt{x}$ to the power of $1$ .

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}} ({\sqrt{x}}^{1} \sqrt{x})}$

Step 2.9.4

Raise $\sqrt{x}$ to the power of $1$ .

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}} ({\sqrt{x}}^{1} {\sqrt{x}}^{1})}$

Step 2.9.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}} {\sqrt{x}}^{1 + 1}}$

Step 2.9.6

Add $1$ and $1$ .

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}} {\sqrt{x}}^{2}}$

Step 2.9.7

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 2.9.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}} {(x^{\frac{1}{2}})}^{2}}$

Step 2.9.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.9.7.3

Combine $\frac{1}{2}$ and $2$ .

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}} x^{\frac{2}{2}}}$

Step 2.9.7.4

Cancel the common factor of $2$ .

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Step 2.9.7.4.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}} x^{\frac{2}{2}}}$

Step 2.9.7.4.2

Rewrite the expression.

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

Step 2.9.7.5

Simplify.

$lim_{x \to \infty} \frac{\sqrt{x}}{e^{\frac{x}{2}} x}$

Step 2.10

Reorder factors in $\frac{\sqrt{x}}{e^{\frac{x}{2}} x}$ .

$lim_{x \to \infty} \frac{\sqrt{x}}{x e^{\frac{x}{2}}}$

Step 3

Reduce.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to \infty} \frac{x^{\frac{1}{2}}}{x e^{\frac{x}{2}}}$

Step 3.2

Factor $x$ out of $x^{\frac{1}{2}}$ .

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

Step 3.3

Cancel the common factors.

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Step 3.3.1

Factor $x$ out of $x e^{\frac{x}{2}}$ .

$lim_{x \to \infty} \frac{x \cdot x^{- \frac{1}{2}}}{x (e^{\frac{x}{2}})}$

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

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

$lim_{x \to \infty} \frac{1}{e^{\frac{x}{2}} x^{\frac{1}{2}}}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{e^{\frac{x}{2}} x^{\frac{1}{2}}}$ approaches $0$ .

$0$