Calculus Examples

$lim_{x \to \infty} \frac{\ln (x)}{\sqrt{x}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to \infty} \ln (x)}{lim_{x \to \infty} \sqrt{x}}$

Step 1.1.2

As log approaches infinity, the value goes to $\infty$ .

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

Step 1.1.3

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

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

Step 1.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 1.2

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.

$lim_{x \to \infty} \frac{\ln (x)}{\sqrt{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [\sqrt{x}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [\sqrt{x}]}$

Step 1.3.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{d}{d x} [\sqrt{x}]}$

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

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

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

Step 1.3.7

Combine the numerators over the common denominator.

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

Step 1.3.8

Simplify the numerator.

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Step 1.3.8.1

Multiply $- 1$ by $2$ .

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

Step 1.3.8.2

Subtract $2$ from $1$ .

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

Step 1.3.9

Move the negative in front of the fraction.

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

Step 1.3.10

Simplify.

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Step 1.3.10.1

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

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

Step 1.3.10.2

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

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

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

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

Step 1.6

Combine factors.

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Step 1.6.1

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

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

Step 1.6.2

Combine $\frac{2}{x}$ and $\sqrt{x}$ .

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

Step 2

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$2 \frac{\infty}{\infty}$

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

$2 \frac{\infty}{\infty}$

Step 3.2

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Combine the numerators over the common denominator.

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

Step 3.3.7

Simplify the numerator.

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Step 3.3.7.1

Multiply $- 1$ by $2$ .

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

Step 3.3.7.2

Subtract $2$ from $1$ .

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

Step 3.3.8

Move the negative in front of the fraction.

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

Step 3.3.9

Simplify.

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Step 3.3.9.1

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

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

Step 3.3.9.2

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

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

Step 3.3.10

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

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

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

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

Step 3.6

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

$1 \cdot 0$

Step 6.2

Multiply $0$ by $1$ .

$0$