Limit of positive infinity plus negative infinity

Question

When trying to evaluate the following limit:

$$\lim_{x\to {\pi \over 2}}\left({1\over x-{\pi \over 2}}+\tan(x)\right)$$

it becomes $\infty^-+\infty^+$. What techniques can be used for solving this?

Answer 1

Set $\dfrac\pi2-x=y,y\to0$

$$\tan x+\dfrac1{x-\dfrac\pi2}=\dfrac1{\tan y}-\dfrac1y=\dfrac{y-\tan y}{y^3}\cdot\dfrac{y^3}{y\tan y}$$

Use Are all limits solvable without L'Hôpital Rule or Series Expansion