Question

$\underset{x\to 0}{lim}\frac{a\sin x-\sin 2x}{{\tan }^{3}x}$ is finite, then $a=$

• A. $0$
• B. $2$
• C. $-2$
• D. $cannotbedetermined$

Answer 1

Answer: (B)

$2$

$\begin{array}{cc}{lim}_{x\to 0}\frac{a\sin x-\sin 2x}{{\tan }^{3}x}isfinite& \left[\sin 2x=2\sin x.\cos x\right]\\ \Rightarrow {lim}_{x\to 0}\frac{\sin x\left(a-2\cos x\right)}{{\sin }^{3}x.{\sec }^{3}x}& \\ \Rightarrow {lim}_{x\to 0}\frac{\left(a-2\cos x\right)}{{\sin }^{2}x.{\sec }^{3}x}& \end{array}$

If is $\frac{0}{0}$ form when $a=2$

Hence, value of $a=2$.