Key to "Determine the one-sided limits numerically or graphically. If..."

Minerva Kline

Answered question

2021-11-12

Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are oo or -oo, and describe the corresponding vertical asymptote.

lim_{x \to 0^{\pm}} \frac{x - \sin | x |}{x^{3}}

Answer & Explanation

Gloria Lusk

Beginner2021-11-13Added 18 answers

To evaluate: $lim_{x \to 0^{+}} \frac{x - \sin | x |}{x^{3}}$
We know that,
$| x | = {\begin{cases} - x & i f & x < 0 \\ x & i f & x \geq 0 \end{cases}$
Therefore,
$lim_{x \to 0^{+}} \frac{x - \sin | x |}{x^{3}} = lim_{x \to 0} \frac{x - \sin (x)}{x^{3}}$
It is $\frac{0}{0}$ form.
Using L-hospital rule.
$lim_{x \to 0} \frac{x - \sin x}{x^{3}} = lim_{x \to 0} \frac{\frac{d}{d x} (x - \sin x)}{\frac{d}{d x} (x^{3})}$
$= lim_{x \to 0} \frac{1 - \cos x}{3 x^{2}}$
$= lim_{x \to 0} \frac{\frac{d}{d x} (1 - \cos x)}{\frac{d}{d x} (3 x^{2})}$
$= lim_{x \to 0} \frac{0 + \sin x}{6 x}$
$= lim_{x \to 0} \frac{\frac{d}{d x} (\sin x)}{\frac{d}{d x} (6 x)}$
$= lim_{x \to 0} \frac{\cos x}{6}$
$= \frac{\cos 0}{6}$
$= \frac{1}{6}$
Hence, required answer is $\frac{1}{6}$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Didn't find what you were looking for?