Question:
Evaluate the following limits:
$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sec ^{2} x-2}{\tan x-1}$
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 Form is $\frac{0}{0}$
By using L hospital Rule,
Differtiate both sides w.r.t $x$
So $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sec ^{2} x-2}{\tan x-1}=$ So $\lim _{x \rightarrow \frac{\pi}{4}} \frac{2 \sec x(\sec x \tan x)-0}{\sec ^{2} x-0}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{2 \sec x(\sec x \tan x)}{\sec ^{2} x}=\lim _{x \rightarrow \frac{\pi}{4}} 2 \tan x=2$
Therefore, $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sec ^{2} x-2}{\tan x-1}=2$