Question

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\tan 2x}{x-{\displaystyle \frac{\mathrm{\pi }}{2}}}$

Answer 1

$$\begin{gathered} \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\tan 2x}{x-{\displaystyle \frac{\mathrm{\pi }}{2}}} \\ =\underset{h\to 0}{\lim }\frac{\tan 2\left({\displaystyle \frac{\mathrm{\pi }}{2}}+h\right)}{{\displaystyle \frac{\mathrm{\pi }}{2}}+h-{\displaystyle \frac{\mathrm{\pi }}{2}}} \\ =\underset{h\to 0}{\lim }\frac{\tan \left(\mathrm{\pi }+2\mathrm{h}\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{2\tan 2h}{2h} \\ \Rightarrow 2 \end{gathered}$$