Question

If $\underset{x\to 0}{\lim }kx\mathrm{cosec}x=\underset{x\to 0}{\lim }x\mathrm{cosec}kx,$ prove that k = ± 1.

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }kx.\mathrm{cosec}x=\underset{x\to 0}{\lim }x\mathrm{cosec}kx \\ \Rightarrow \underset{x\to 0}{\lim }\left[\frac{kx}{\sin x}\right]=\underset{x\to 0}{\lim }\left[\frac{x}{\sin \left(kx\right)}\right] \\ \Rightarrow k\underset{x\to 0}{\lim }\left[\frac{x}{\sin x}\right]=\underset{x\to 0}{\lim }\left[\frac{kx}{\sin \left(kx\right)}\right]\times \frac{1}{k} \\ \Rightarrow k=\frac{1}{k} \\ \Rightarrow k^2=1 \\ \Rightarrow k=\pm 1 \end{gathered}$$