Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

Expand $\ln (x^{\frac{1}{\ln (x)}})$ by moving $\frac{1}{\ln (x)}$ outside the logarithm.

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

Simplify the limit argument.

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor of $\ln (x)$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

$e^{lim_{x \to \infty} 1}$

Step 2.3

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

$e^{1}$

Step 3

Simplify.

$e$

Step 4

The result can be shown in multiple forms.

Exact Form:

$e$

Decimal Form:

$2.71828182 \dots$