Question:
Evaluate the following limits:
$\lim _{x \rightarrow \frac{\pi}{4}} \frac{1-\tan x}{x-\frac{\pi}{4}}$
Solution:
To Find: Limits
NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form
In this Case, indeterminate Formis $\frac{0}{0}$
By using $L$ hospital Rule,
Differtiate both sides w.r.t $x$
So $\lim _{x \rightarrow \frac{\pi}{4}} \frac{1-\tan x}{x-\frac{\pi}{4}}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{0-\sec ^{2} x}{1-0}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{-\sec ^{2} x}{1}=-2$
Therefore, $\lim _{x \rightarrow \frac{\pi}{4}} \frac{1-\tan x}{x-\frac{\pi}{4}}=-2$