Question

Given $l_{1}=\lim _{x \rightarrow \frac{\pi}{4}} \cos ^{-1}\left[\sec \left(x-\frac{\pi}{4}\right)\right]$; $l_{2}=\lim _{x \rightarrow \frac{\pi}{4}} \sin ^{-1}\left[\operatorname{cosec}\left(x+\frac{\pi}{4}\right)\right]$ $l_{3}=\lim _{x \rightarrow \frac{\pi}{4}} \tan ^{-1}\left[\cot \left(x+\frac{\pi}{4}\right)\right]$ $l_{4}=\lim _{x \rightarrow \frac{\pi}{4}} \cot ^{-1}\left[\tan \left(x-\frac{\pi}{4}\right)\right]$ where \([\mathrm{x}]\) denotes greatest integer function then which of the following limits exist (A) \(l_{1}\) (B) \(l_{2}\) (D) \(l_{4}\) (C) \(l_{3}\)

Short answer

a) \(l_{1}\) b) \(l_{2}\) c) \(l_{3}\) d) \(l_{4}\) Answer: All of the given limits, \(l_1, l_2, l_3,\) and \(l_4\), exist.

Step-by-step solution

Compute limit \(l_{1}\)

To compute the limit \(l_{1}\), we have $$l_{1} = \lim _{x \rightarrow \frac{\pi}{4}} \cos ^{-1}\left[\sec\left(x-\frac{\pi}{4}\right)\right].$$ Using the identities: \(\cos^{-1}(x) = \pi/2 - \sin^{-1}(x)\) and \(\sec(x) = 1/\cos(x)\), we can modify the limit as $$l_1 = \lim_{x\rightarrow\frac{\pi}{4}}\left(\frac{\pi}{2} - \sin^{-1}\left(\cos\left(x-\frac{\pi}{4}\right)\right)\right).$$ Now evaluate the limit, using the fact that \(\sin^{-1}(\sin(x)) = x\), $$l_1 = \frac{\pi}{2} - \left(\frac{\pi}{4} - \frac{\pi}{4}\right) = \frac{\pi}{2}.$$ So, the limit \(l_{1}\) exists.

Compute limit \(l_{2}\)

To compute the limit \(l_{2}\), we have $$l_{2} = \lim _{x \rightarrow \frac{\pi}{4}} \sin ^{-1}\left[\operatorname{cosec}\left(x+\frac{\pi}{4}\right)\right].$$Using the identity: \(\cosec(x) = 1/\sin(x)\), we can modify the limit as $$l_2 = \lim_{x\rightarrow\frac{\pi}{4}} \sin^{-1}\left(\frac{1}{\sin\left(x +\frac{\pi}{4}\right)}\right).$$ Now evaluate the limit, using the fact that \(\sin^{-1}(\sin(x)) = x\), $$l_2 = \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2}.$$ So, the limit \(l_{2}\) exists.

Compute limit \(l_{3}\)

To compute the limit \(l_{3}\), we have $$l_{3}=\lim _{x \rightarrow \frac{\pi}{4}} \tan ^{-1}\left[\cot \left(x+\frac{\pi}{4}\right)\right].$$ Using the identities: \(\tan^{-1}(x) = \pi/2 - \cot^{-1}(x)\) and \(\cot(x) = 1/\tan(x)\), we can modify the limit as $$l_3 = \lim_{x\rightarrow\frac{\pi}{4}}\left(\frac{\pi}{2} - \cot^{-1}\left(\tan\left(x+\frac{\pi}{4}\right)\right)\right).$$ Now evaluate the limit, using the fact that \(\cot^{-1}(\cot(x)) = x\), $$l_3 = \frac{\pi}{2} - \left(\frac{\pi}{4} + \frac{\pi}{4}\right) = 0.$$ So, the limit \(l_{3}\) exists.

Compute limit \(l_{4}\)

To compute the limit \(l_{4}\), we have $$l_{4}=\lim _{x \rightarrow \frac{\pi}{4}} \cot ^{-1}\left[\tan \left(x-\frac{\pi}{4}\right)\right].$$ Using the identities: \(\cot^{-1}(x) = \pi/2 - \tan^{-1}(x)\) and \(\tan(x) = 1/\cot(x)\), we can modify the limit as $$l_4 = \lim_{x\rightarrow\frac{\pi}{4}}\left(\frac{\pi}{2} - \tan^{-1}\left(\cot\left(x-\frac{\pi}{4}\right)\right)\right).$$ Now evaluate the limit, using the fact that \(\tan^{-1}(\tan(x)) = x\), $$l_4 = \frac{\pi}{2} - \left(\frac{\pi}{4} - \frac{\pi}{4}\right) = \frac{\pi}{2}.$$ So, the limit \(l_{4}\) exists. Now, all the limits \(l_1, l_2, l_3,\) and \(l_4\) exist. Therefore, the correct answer is all options (A), (B), (C), and (D).