Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 3.1

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

Tap for more steps...

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.

Tap for more steps...

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 $g (x) = \ln (x)$ .

Tap for more steps...

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 $1$ .

$1$