Question

The value of $\underset{x\to 0}{lim}\left(\frac{x}{\sqrt[8]{81-\sin x}-\sqrt[8]{81+\sin x}}\right)$ is equal to

• A. $4$
• B. $-4$
• C. $-1$
• D. $0$

Answer 1

The correct option is B $-4$

Let consider

$L=\underset{x\to 0}{lim}\frac{\sqrt[8]{81-\sin x}-\sqrt[8]{81+\sin x}}{x}=\underset{x\to 0}{lim}\frac{\sqrt[8]{81-\sin x}-\sqrt[8]{81+\sin x}}{x}\times \frac{\sin x}{\sin x}=\underset{x\to 0}{lim}\frac{\sqrt[8]{81-\sin x}-\sqrt[8]{81+\sin x}}{\sin x}=\underset{x\to 0}{lim}\frac{\sqrt[8]{81-\sin x}-\sqrt[8]{81+\sin x}-1+1}{\sin x}=\underset{x\to 0}{lim}(\frac{\sqrt[8]{81-\sin x}-1}{\sin x}-\frac{\sqrt[8]{81+\sin x}-1}{\sin x})=\underset{x\to 0}{lim}\left(\right(-1)\frac{(1-\sin x{)}^{1/8}-1^{1/8}}{(1-\sin x)-1}-\frac{(1+\sin x{)}^{1/8}-1^{1/8}}{(1+\sin x)-1})=(-1)\frac{1}{8}\cdot 1-\frac{1}{8}=-\frac{1}{4}$

Now, $\frac{1}{L}=\underset{x\to 0}{lim}\left(\frac{x}{\sqrt[8]{81-\sin x}-\sqrt[8]{81+\sin x}}\right)=-4$