Calculus Examples

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

Step 1

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

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

Step 2

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

$- \infty - lim_{x \to 0} x^{- 2}$

Step 3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \infty - lim_{x \to 0} \frac{1}{x^{2}}$

Step 4

Since the numerator is a constant and the denominator approaches $0$ when $x$ approaches $0$ , the fraction $\frac{1}{x^{2}}$ approaches infinity.

$- \infty - \infty$

Step 5

Simplify the answer.

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Step 5.1

A non-zero constant times infinity is infinity.

$- \infty + - \infty$

Step 5.2

Infinity plus infinity is infinity.

$- \infty$