Calculus Examples

$lim_{x \to 0} \frac{\ln ({(x)}^{2})}{x - 1}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \ln ({(x)}^{2})}{lim_{x \to 0} x - 1}$

Step 2

As $x$ approaches $0$ from either side, $\ln ({(x)}^{2})$ decreases without bound.

$\frac{- \infty}{lim_{x \to 0} x - 1}$

Step 3

Evaluate the limit.

Tap for more steps...

Step 3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{- \infty}{lim_{x \to 0} x - lim_{x \to 0} 1}$

Step 3.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{- \infty}{lim_{x \to 0} x - 1 \cdot 1}$

Step 4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- \infty}{0 - 1 \cdot 1}$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Simplify the denominator.

Tap for more steps...

Step 5.1.1

Multiply $- 1$ by $1$ .

$\frac{- \infty}{0 - 1}$

Step 5.1.2

Subtract $1$ from $0$ .

$\frac{- \infty}{- 1}$

Step 5.2

Move the negative one from the denominator of $\frac{- \infty}{- 1}$ .

$- 1 \cdot - \infty$

Step 5.3

Rewrite $- 1 \cdot - \infty$ as $- - \infty$ .

$- - \infty$

Step 5.4

A non-zero constant times infinity is infinity.

$\infty$