Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

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

Step 3.7.2

Subtract $2$ from $1$ .

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

Step 3.8

Move the negative in front of the fraction.

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

Step 3.9

Simplify.

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Step 3.9.1

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

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

Step 3.9.2

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

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

Step 3.10

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Evaluate the limit.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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

$\frac{1}{2} \cdot 0$

Step 8

Multiply $\frac{1}{2}$ by $0$ .

$0$