Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite

${(\ln (x))}^{\frac{1}{x}}$ as

$e^{\ln ({(\ln (x))}^{\frac{1}{x}})}$ .

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

Step 1.2

Expand

$\ln ({(\ln (x))}^{\frac{1}{x}})$ by moving

$\frac{1}{x}$ outside the logarithm.

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

Combine

$\frac{1}{x}$ and

$\ln (\ln (x))$ .

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

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.

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

Step 3.1.2

As log approaches infinity, the value goes to

$\infty$ .

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

Step 3.1.3

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

$e^{\frac{\infty}{\infty}}$

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{\infty}{\infty}}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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 3.3.2.1

To apply the Chain Rule, set

$u$ as

$\ln (x)$ .

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

Step 3.3.2.2

The derivative of

$\ln (u)$ with respect to

$u$ is

$\frac{1}{u}$ .

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

Step 3.3.2.3

Replace all occurrences of

$u$ with

$\ln (x)$ .

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

Step 3.3.3

The derivative of

$\ln (x)$ with respect to

$x$ is

$\frac{1}{x}$ .

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

Step 3.3.4

Multiply

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

$\frac{1}{x}$ .

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

Step 3.3.5

Reorder terms.

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

Step 3.3.6

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Multiply

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

$1$ .

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

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction

$\frac{1}{x \ln (x)}$ approaches

$0$ .

$e^{0}$

Step 5

Anything raised to

$0$ is

$1$ .

$1$