Evaluate the following limits:
Question:

Evaluate the following limits:

$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\tan ^{3} x-\tan x}{\cos \left(x+\frac{\pi}{4}\right)}$

 

Solution:

$=\lim _{x \rightarrow \frac{\pi}{4}} \frac{\left(\frac{-\sin x(\cos 2 x)}{\cos x \cos x \cos x}\right)}{\left(\frac{\cos x-\sin x}{\sqrt{2}}\right)}$

$=-\sqrt{2} \times \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x(\cos x+\sin x)}{\cos x \times \cos x \times \cos x}$

$=-\sqrt{2} \times \frac{\frac{1}{\sqrt{2}} \times \sqrt{2}}{\frac{1}{\sqrt{2}}^{3}}$

$=-4$

 

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