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

$x$ 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.

$- \infty$