Q. If for some real number $k$ $\lim_{x \rightarrow 0}{kx\ cosec\ (x) = \lim_{x \rightarrow 0}{x\ cosec\ (kx)}}$, then the possible values of $k$ are
A) 1, -1
B) 0, 1
C) 1, 2
D) $0,\ \pi$
Explanation
(a)
Given, $\lim_{x \rightarrow 0}{kx\ cosec\ (x) = \lim_{x \rightarrow 0}{\ x\ cosec\ (kx)}}$
$\Rightarrow \ \ \ k\lim_{x \rightarrow 0}{\frac{x}{\sin x} =}\lim_{x \rightarrow 0}\frac{x}{\sin{kx}} \times \frac{k}{k}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ k = \frac{1}{k}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ k = \pm 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.