Calculus Examples

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

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 $0$ .

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

$0^{2} - - \infty$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raising $0$ to any positive power yields $0$ .

$0 - - \infty$

Step 4.1.2

A non-zero constant times infinity is infinity.

$0 + \infty$

Step 4.2

Add $0$ and $\infty$ .

$\infty$