Calculus Examples

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

Step 1

Move the limit inside the logarithm.

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

Step 2

Use the properties of logarithms to simplify the limit.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 3

Evaluate the limit.

Tap for more steps...

Step 3.1

Move the limit into the exponent.

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

Step 3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 3.4

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

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

Step 3.5

Move the limit inside the logarithm.

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

Step 4

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

Tap for more steps...

Step 4.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 4.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\ln (e^{(1 - 1 \cdot 1) \cdot \ln (1)})$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Use logarithm rules to move $(1 - 1 \cdot 1) \cdot \ln (1)$ out of the exponent.

$(1 - 1 \cdot 1) \cdot \ln (1) \ln (e)$

Step 5.2

Multiply $- 1$ by $1$ .

$(1 - 1) \cdot \ln (1) \ln (e)$

Step 5.3

Subtract $1$ from $1$ .

$0 \cdot \ln (1) \ln (e)$

Step 5.4

The natural logarithm of $1$ is $0$ .

$0 \cdot 0 \ln (e)$

Step 5.5

Multiply $0$ by $0$ .

$0 \ln (e)$

Step 5.6

The natural logarithm of $e$ is $1$ .

$0 \cdot 1$

Step 5.7

Multiply $0$ by $1$ .

$0$