Calculus Examples

$lim_{x \to 0} x^{3} - \frac{1}{x^{2}}$

Step 1

Change the two-sided limit into a left sided limit.

$lim_{x \to 0^{-}} x^{3} - \frac{1}{x^{2}}$

Step 2

Evaluate the limit.

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Step 2.1

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

$lim_{x \to 0^{-}} x^{3} - lim_{x \to 0^{-}} \frac{1}{x^{2}}$

Step 2.2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 0^{-}} x)}^{3} - lim_{x \to 0^{-}} \frac{1}{x^{2}}$

Step 3

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

${(lim_{x \to 0^{-}} x)}^{3} - \infty$

Step 4

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

$0^{3} - \infty$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

$0 - \infty$

Step 5.1.2

A non-zero constant times infinity is infinity.

$0 + - \infty$

Step 5.2

Subtract $\infty$ from $0$ .

$- \infty$