Evaluate the following limits:

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, $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sec ^{2} x-2}{\tan x-1}=2$

 

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