Calculus Examples

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

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.

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

Step 1.2

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

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

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Reorder terms.

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

Step 3.6

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Simplify terms.

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Step 5.1

Combine $\frac{1}{x \ln (x)}$ and $x$ .

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

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 6

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

$0$