Question

Evaluate: $\underset{x\to 0}{lim}\frac{\sqrt[3]{31+\sin x}-\sqrt[3]{31-\sin x}}{x}$

• A. $0$
• B. $1$
• C. $\frac{2}{3}$
• D. $\frac{3}{2}$

Answer 1

The correct option is C $\frac{2}{3}$

Rationalize the numerator by multiplying and dividing with
${(1+\sin x)}^{\frac{2}{3}}+{(1-\sin x)}^{\frac{2}{3}}+{(1-{\sin }^{2}x)}^{\frac{1}{3}}$, we get,
$=\frac{2\sin x}{x({(1+\sin x)}^{\frac{2}{3}}+{(1-\sin x)}^{\frac{2}{3}}+{(1-{\sin }^{2}x)}^{\frac{2}{3}})}$
Now substituting the limit as $x\to 0$, we get answer as $\frac{2}{3}$