Calculus Examples

$lim_{x \to 8} {(\ln (x))}^{\frac{1}{x}}$

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

$lim_{x \to 8} e^{\ln ({(\ln (x))}^{\frac{1}{x}})}$

Step 1.2

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

$lim_{x \to 8} e^{\frac{1}{x} \ln (\ln (x))}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 8} \frac{1}{x} \ln (\ln (x))}$

Step 2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$e^{lim_{x \to 8} \frac{1}{x} \cdot lim_{x \to 8} \ln (\ln (x))}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$e^{\frac{lim_{x \to 8} 1}{lim_{x \to 8} x} \cdot lim_{x \to 8} \ln (\ln (x))}$

Step 2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$e^{\frac{1}{lim_{x \to 8} x} \cdot lim_{x \to 8} \ln (\ln (x))}$

Step 2.5

Move the limit inside the logarithm.

$e^{\frac{1}{lim_{x \to 8} x} \cdot \ln (lim_{x \to 8} \ln (x))}$

Step 2.6

Move the limit inside the logarithm.

$e^{\frac{1}{lim_{x \to 8} x} \cdot \ln (\ln (lim_{x \to 8} x))}$

Step 3

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

Tap for more steps...

Step 3.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$e^{\frac{1}{8} \cdot \ln (\ln (lim_{x \to 8} x))}$

Step 3.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$e^{\frac{1}{8} \cdot \ln (\ln (8))}$

Step 4

Combine $\frac{1}{8}$ and $\ln (\ln (8))$ .

$e^{\frac{\ln (\ln (8))}{8}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{\ln (\ln (8))}{8}}$

Decimal Form:

$1.09583038 \dots$