Limit with ln(tan x)

Question

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

Answer 1

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 $a,$ $$\lim_{h\to0}\dfrac{\ln(1+a\tanh)}h=a\lim_{h\to0}\dfrac{\ln(1+a\tan h)}{a\tan h}\cdot\dfrac{\tan h}h=\cdots =a$$

Answer 2

Let $f(x) = \ln (\tan x).$ We are looking at

$$ \frac{f(x)}{x-\pi/4} =\frac{f(x) - f(\pi/4)}{x-\pi/4}.$$

By definition the limit of this as $x\to \pi/4$ is just $f'(\pi/4),$ an easy computation.