Question

The value of $\lim_{x \to 0} \left(\frac{x}{\sqrt[8]{1-\sin x} - \sqrt[8]{1+\sin x}}\right)$ is equal to :

• A. 0
• B. -1
• C. -4
• D. 4

Hint:

For limits involving roots and trigonometric functions as $x \to 0$, the binomial approximation $(1+x)^n \approx 1+nx$ and the standard limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$ are powerful tools that are often faster than applying L'Hopital's Rule.

Answer 1

The Correct Option is C

Consider: \[ \lim_{x\to0}\frac{x}{(1-\sin x)^{1/8}-(1+\sin x)^{1/8}} \] Using binomial approximation for small $u$: \[ (1+u)^{1/8}\approx1+\frac{u}{8} \] \[ (1-\sin x)^{1/8}\approx1-\frac{\sin x}{8},\quad (1+\sin x)^{1/8}\approx1+\frac{\sin x}{8} \] Denominator: \[ -\frac{\sin x}{4} \] Thus, \[ \lim_{x\to0}\frac{x}{-\frac{\sin x}{4}} = -4\lim_{x\to0}\frac{x}{\sin x} \] Since $\lim_{x\to0}\frac{\sin x}{x}=1$, \[ \boxed{-4} \]