Q103 / 160 · TS EAMCET 2023 (14 May Shift 2) · Mathematics

If a function $f$ is defined by $f(x)=\frac{\cot ^3 x-\tan x}{\cos (x+\pi / 4)}(x \neq \pi / 4)$, then $\lim _{x \rightarrow \frac{\pi}{4}} f(x)=$

  • A
    4
  • Correct answer
    B
    8
  • C
    $\frac{8}{3}$
  • D
    16

Solution

$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\cot ^3 x-\tan x}{\cos \left(x+\frac{\pi}{4}\right)}$ $\begin{aligned} & =\lim _{x \rightarrow \frac{\pi}{4}} \frac{\frac{1}{\tan 3}-\tan x}{\cos x \cos \frac{\pi}{4}-\sin x \sin \frac{\pi}{4}} \\ & =\lim _{x \rightarrow \frac{\pi}{4}} \frac{1-\tan ^4 x}{\tan ^3 x \cdot \frac{1}{\sqrt{2}}(\cos x-\sin x)} \\ & =\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2}\left(1+\tan ^2 x\right)(1+\tan x)(1-\tan x)}{\tan ^2 x \cdot \tan x \cdot \cos x(1-\tan x)} \\ & =\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} \sec ^2 x(1+\tan x)}{\sin ^2 x \cdot \tan ^2 x} \\ & =2(\sqrt{2})^2(1+1)=8 .\end{aligned}$

Hence, the correct answer is option: 'B'.

Relevant Questions