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Evaluate the following limits:

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

 

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