Calculus Examples

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

Simplify the limit argument.

Tap for more steps...

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

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

Combine factors.

Tap for more steps...

Combine $- 2$ and $\frac{1}{x^{3}}$ .

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

Combine $\frac{- 2}{x^{3}}$ and $\sin (x)$ .

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

Apply L'Hospital's rule.

Tap for more steps...

Evaluate the limit of the numerator and the limit of the denominator.

Tap for more steps...

Take the limit of the numerator and the limit of the denominator.

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

Evaluate the limit of the numerator.

Tap for more steps...

Evaluate the limit.

Tap for more steps...

Move the term $- 2$ outside of the limit because it is constant with respect to $x$ .

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

Move the limit inside the trig function because sine is continuous.

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

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

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

Simplify the answer.

Tap for more steps...

The exact value of $\sin (0)$ is $0$ .

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

Multiply $- 2$ by $0$ .

$\frac{0}{lim_{x \to 0} x^{3}}$

Evaluate the limit of the denominator.

Tap for more steps...

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

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

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

$\frac{0}{0^{3}}$

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

$\frac{0}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{- 2 \sin (x)}{x^{3}} = lim_{x \to 0} \frac{\frac{d}{d x} [- 2 \sin (x)]}{\frac{d}{d x} [x^{3}]}$

Find the derivative of the numerator and denominator.

Tap for more steps...

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [- 2 \sin (x)]}{\frac{d}{d x} [x^{3}]}$

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 \sin (x)$ with respect to $x$ is $- 2 \frac{d}{d x} [\sin (x)]$ .

$lim_{x \to 0} \frac{- 2 \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x^{3}]}$

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$lim_{x \to 0} \frac{- 2 \cos (x)}{\frac{d}{d x} [x^{3}]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Since the numerator is negative and the denominator $3 x^{2}$ approaches zero and is greater than zero for $x$ near $0$ on both sides, the function decreases without bound.

$- \infty$