Q:

Limit with ln(tan x)

$$\lim_{x\to\pi/4} \frac{\ln(\tan(x))}{x-\pi/4}$$
Could you help me finding the limit? I tried some trigonometrical conversions but got stucked. 

A:

Set $\dfrac\pi4-x=h$ to get
$$-\lim_{h\to0}\dfrac{\ln\tan\left(\dfrac\pi4-h\right)}h =\lim_{h\to0}\dfrac{\ln(1+\tan h)}h-\lim_{h\to0}\dfrac{\ln(1-\tanh)}h$$
Now any constant $a,$ $$\lim_{h\to0}\dfrac{\ln(1+a\tanh)}h=a\lim_{h\to0}\dfrac{\ln(1+a\tan h)}{a\tan h}\cdot\dfrac{\tan h}h=\cdots =a$$

