Question

Write the value of $\underset{x\to \mathrm{\pi }/2}{\lim }\frac{2x-\mathrm{\pi }}{\cos x}.$

Answer 1

$$\begin{gathered} \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left[\frac{2x-\mathrm{\pi }}{\cos x}\right] \\ \mathrm{LHL}: \\ \underset{x\to {\frac{\mathrm{\pi }}{2}}^{-}}{\lim }\left[\frac{2x-\mathrm{\pi }}{\cos x}\right] \\ \mathrm{Let}x=\frac{\mathrm{\pi }}{2}-h \\ \mathrm{If}x\to \frac{\mathrm{\pi }}{2},\mathrm{then}\mathrm{we}\mathrm{have}: \\ h\to 0 \\ =\underset{h\to 0}{\lim }\left[\frac{2\left({\displaystyle \frac{\mathrm{\pi }}{2}}-h\right)-\mathrm{\pi }}{\cos \left({\displaystyle \frac{\mathrm{\pi }}{2}}-h\right)}\right] \\ =\underset{h\to 0}{\lim }\left[\frac{\mathrm{\pi }-2h-\mathrm{\pi }}{\sin h}\right] \\ =-2 \\ \mathrm{RHL}: \\ \underset{x\to {\frac{\mathrm{\pi }}{2}}^{+}}{\lim }\left[\frac{2x-\mathrm{\pi }}{\cos x}\right] \\ \mathrm{Let}x=\frac{\mathrm{\pi }}{2}+h \\ \mathrm{If}x\to \frac{\mathrm{\pi }}{2},\mathrm{then}\mathrm{we}\mathrm{have}: \\ h\to 0 \\ =\underset{h\to 0}{\lim }\left[\frac{2\left({\displaystyle \frac{\mathrm{\pi }}{2}}+h\right)-\mathrm{\pi }}{\cos \left({\displaystyle \frac{\mathrm{\pi }}{2}}+h\right)}\right] \\ =\underset{h\to 0}{\lim }\left[\frac{2h}{-\sin h}\right] \\ =-2 \\ \Rightarrow \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left(\frac{2x-\mathrm{\pi }}{\cos x}\right)=-2 \end{gathered}$$