Limit Continuity and Differentiability 1 Question 4
5. $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\cot ^{3} x-\tan x}{\cos x+\frac{\pi}{4}}$ is
(2019 Main, 12 Jan I)
(a) $4 \sqrt{2}$
(b) 4
(c) 8
(d) $8 \sqrt{2}$
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Answer:
Correct Answer: 5. (c)
Solution:
- $\lim _{x \rightarrow 0^{-}} \frac{x([x]+|x|) \sin [x]}{|x|}=\lim _{x \rightarrow 0^{-}} \frac{x([x]-x) \sin [x]}{-x}$
$$ \begin{array}{lr} =\lim _{x \rightarrow 0^{-}} \frac{x(-1-x) \sin (-1)}{-x} \quad(\because|x|=-x, \text { if } x<0) \\ =\lim _{x \rightarrow 0^{-}} \frac{-x(x+1) \sin (-1)}{-x}=\lim _{x \rightarrow 0^{-}}(x+1) \sin (-1) \\ =(0+1) \sin (-1)(\text { by direct substitution }) \\ =-\sin 1 \quad(\because \lim (-\theta)=-\sin \theta) \end{array} $$
