Calculus Examples

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

Step 1

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

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

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

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

Step 3.2

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

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

Step 3.3

Move the limit inside the logarithm.

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

Step 3.4

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Simplify $1 \cdot \ln (1)$ by moving $1$ inside the logarithm.

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

Step 5.1.2

Exponentiation and log are inverse functions.

$\frac{1^{1} - 1 \cdot 1}{\ln (x)}$

Step 5.1.3

Evaluate the exponent.

$\frac{1 - 1 \cdot 1}{\ln (x)}$

Step 5.1.4

Multiply $- 1$ by $1$ .

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

Step 5.1.5

Subtract $1$ from $1$ .

$\frac{0}{\ln (x)}$

Step 5.2

Divide $0$ by $\ln (x)$ .

$0$