Evaluate the follwing trig limit?

Question

lim x---> pi/4   (tanx - cotx) / (x -pi/4)

Answer 1

The limit is 4. Here’s the reason:

Let x=π/4+h where h is small so that tan(h)≈h.

tan(x)=(tan(π/4)+tan(h))/(1-tan(π/4)tan(h))=(1+tan(h))/(1-tan(h))

cot(x)=(1-tan(h))/(1+tan(h))

tan(x)-cot(x)=((1+tan(h))²-(1-tan(h))²)/(1-tan²(h))=

4tan(h)/(1-tan²(h)≈4h/(1-h²)

So limit x→π/4 (tan(x)-cot(x))/(x-π/4) is limit h→0 (4h/(1-h²))/h=4/(1-h²)→4.