Calculus Examples

$\frac{lim_{x \to 1} x^{2} - 1}{\ln (x^{2})}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 1} x^{2} - lim_{x \to 1} 1}{\ln (x^{2})}$

Step 1.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 1}{\ln (x^{2})}$

Step 1.3

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

$\frac{{(lim_{x \to 1} x)}^{2} - 1 \cdot 1}{\ln (x^{2})}$

Step 2

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

$\frac{1^{2} - 1 \cdot 1}{\ln (x^{2})}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

$\frac{1 - 1 \cdot 1}{\ln (x^{2})}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{\ln (x^{2})}$

Step 3.1.3

Subtract $1$ from $1$ .

$\frac{0}{\ln (x^{2})}$

Step 3.2

Expand $\ln (x^{2})$ by moving $2$ outside the logarithm.

$\frac{0}{2 \ln (x)}$

Step 3.3

Cancel the common factor of $0$ and $2$ .

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Step 3.3.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 \ln (x)}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $2$ out of $2 \ln (x)$ .

$\frac{2 (0)}{2 (\ln (x))}$

Step 3.3.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 \ln (x)}$

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

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

$0$