Calculus Examples

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

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.

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

Step 1.2

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

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

Step 1.3

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

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

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

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

Step 3.7.2

Subtract $2$ from $1$ .

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

Step 3.8

Move the negative in front of the fraction.

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

Step 3.9

Simplify.

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Step 3.9.1

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

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

Step 3.9.2

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

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

Step 3.10

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

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Step 3.10.1

To apply the Chain Rule, set $u$ as $\ln (x)$ .

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

Step 3.10.2

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

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

Step 3.10.3

Replace all occurrences of $u$ with $\ln (x)$ .

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

Step 3.11

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

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

Step 3.12

Multiply $\frac{1}{\ln (x)}$ by $\frac{1}{x}$ .

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

Step 3.13

Reorder terms.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Combine factors.

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Step 6.1

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

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

Step 6.2

Combine $\frac{x}{2 \sqrt{x}}$ and $\ln (x)$ .

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

Step 7

Apply L'Hospital's rule.

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

Evaluate the limit of the numerator.

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Step 7.1.2.1

Evaluate the limit.

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Step 7.1.2.1.1

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

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

Step 7.1.2.1.2

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

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

Step 7.1.2.2

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

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

Step 7.1.2.3

Infinity times infinity is infinity.

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

Step 7.1.3

Since the function $\sqrt{x}$ approaches $\infty$ , the positive constant $2$ times the function also approaches $\infty$ .

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Step 7.1.3.1

Consider the limit with the constant multiple $2$ removed.

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

Step 7.1.3.2

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

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

Step 7.1.3.3

Infinity divided by infinity is undefined.

Undefined

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

Step 7.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 7.2

$lim_{x \to \infty} \frac{x \ln (x)}{2 \sqrt{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x \ln (x)]}{\frac{d}{d x} [2 \sqrt{x}]}$

Step 7.3

Find the derivative of the numerator and denominator.

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Step 7.3.1

Differentiate the numerator and denominator.

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

Step 7.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x)$ .

$lim_{x \to \infty} \frac{x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x]}{\frac{d}{d x} [2 \sqrt{x}]}$

Step 7.3.3

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

$lim_{x \to \infty} \frac{x \frac{1}{x} + \ln (x) \frac{d}{d x} [x]}{\frac{d}{d x} [2 \sqrt{x}]}$

Step 7.3.4

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

$lim_{x \to \infty} \frac{\frac{x}{x} + \ln (x) \frac{d}{d x} [x]}{\frac{d}{d x} [2 \sqrt{x}]}$

Step 7.3.5

Cancel the common factor of $x$ .

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Step 7.3.5.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x}{x} + \ln (x) \frac{d}{d x} [x]}{\frac{d}{d x} [2 \sqrt{x}]}$

Step 7.3.5.2

Rewrite the expression.

$lim_{x \to \infty} \frac{1 + \ln (x) \frac{d}{d x} [x]}{\frac{d}{d x} [2 \sqrt{x}]}$

Step 7.3.6

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

$lim_{x \to \infty} \frac{1 + \ln (x) \cdot 1}{\frac{d}{d x} [2 \sqrt{x}]}$

Step 7.3.7

Multiply $\ln (x)$ by $1$ .

$lim_{x \to \infty} \frac{1 + \ln (x)}{\frac{d}{d x} [2 \sqrt{x}]}$

Step 7.3.8

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

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

Step 7.3.9

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

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

Step 7.3.10

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

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

Step 7.3.11

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

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

Step 7.3.12

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

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

Step 7.3.13

Combine the numerators over the common denominator.

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

Step 7.3.14

Simplify the numerator.

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Step 7.3.14.1

Multiply $- 1$ by $2$ .

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

Step 7.3.14.2

Subtract $2$ from $1$ .

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

Step 7.3.15

Move the negative in front of the fraction.

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

Step 7.3.16

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

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

Step 7.3.17

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

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

Step 7.3.18

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

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

Step 7.3.19

Cancel the common factor.

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

Step 7.3.20

Rewrite the expression.

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

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.5

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

$lim_{x \to \infty} (1 + \ln (x)) \sqrt{x}$

Step 8

Evaluate the limit.

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Step 8.1

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

$lim_{x \to \infty} 1 + \ln (x) \cdot lim_{x \to \infty} \sqrt{x}$

Step 8.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$(lim_{x \to \infty} 1 + lim_{x \to \infty} \ln (x)) \cdot lim_{x \to \infty} \sqrt{x}$

Step 8.3

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

$(1 + lim_{x \to \infty} \ln (x)) \cdot lim_{x \to \infty} \sqrt{x}$

Step 9

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

$(1 + \infty) \cdot lim_{x \to \infty} \sqrt{x}$

Step 10

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

$(1 + \infty) \cdot \infty$

Step 11

Simplify the answer.

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Step 11.1

Infinity plus or minus a number is infinity.

$\infty \cdot \infty$

Step 11.2

Infinity times infinity is infinity.

$\infty$