Question

Consider the function \(f(x)=\cos ^{-1}[\cot x]\) where [ ] indicates greatest integer function. Assertion (A): \(\lim _{x \rightarrow \frac{\pi}{2}} f(x)\) exists Reason \((\mathbf{R}):\) Both \(\lim \mathrm{f}(\mathrm{x})\) and $\lim \mathrm{f}(\mathrm{x})$ are finite. \(x \rightarrow \frac{\pi}{2} \quad x \rightarrow \frac{\pi}{2}\)

Short answer

In this problem, we are given a function \(f(x)=\cos ^{-1}[\cot x]\) and we need to find the limit of this function as x approaches π/2. After analyzing the properties of the cosine inverse and greatest integer functions, and computing the left and right limits as x approaches π/2, we found that the limit does not exist. Thus, Assertion (A) is false, and Reason (R) is irrelevant in this case. Answer: Assertion (A) is false and Reason (R) is irrelevant.

Step-by-step solution

Understand the given function

The given function is \(f(x)=\cos ^{-1}[\cot x]\). Here, \(\cos^{-1}\) means the inverse of the cosine function, and \([\cdot]\) denotes the greatest integer function, which gives the largest integer less than or equal to a given number. The domain of \(\cos ^{-1}\) is \([-1, 1]\), so cot(x) must be in the range \([-1,1]\) for the function to have a value.

Simplify f(x)

f(x) can be written as \(\cos^{-1}[\cot x]=\cos^{-1}\left[\frac{\cos x}{\sin x}\right]\). As we are interested in the limit as x approaches π/2, let's find out the values of sin(x) and cos(x) as x \(\rightarrow \frac{\pi}{2}\). As \(x\rightarrow \frac{\pi}{2}\), \(\sin x \rightarrow 1\) and \(\cos x \rightarrow 0\). Now, we need to find the values for the greatest integer \(\left[\frac{\cos x}{\sin x}\right]\) as \(x\rightarrow \frac{\pi}{2}\).

Compute limits for sine and cosine functions separately

Let's investigate the limits as \(x \rightarrow \frac{\pi}{2}^-\) and \(x\rightarrow \frac{\pi}{2}^+\), this means finding the left limit and right limit of \(\left[\frac{\cos x}{\sin x}\right]\) as x approaches π/2. Left limit (x \(\rightarrow \frac{\pi}{2}^-\)): \(\lim_{x \rightarrow \frac{\pi}{2}^{-}} \left[\frac{\cos x}{\sin x}\right]=\left[\frac{0^+}{1}\right]=0\) Right limit (x \(\rightarrow \frac{\pi}{2}^+\)): \(\lim_{x \rightarrow \frac{\pi}{2}^{+}} \left[\frac{\cos x}{\sin x}\right]=\left[\frac{0^-}{1}\right]=-1\).

Determine if the limit exists

Now that we have the left limit and right limit of \(\left[\frac{\cos x}{\sin x}\right]\) as x approaches π/2, we can check whether the limit exists or not. For the limit to exist, both left and right limits must be equal. Since the left limit (0) and right limit (-1) are not equal, the limit of the greatest integer function inside the cosine inverse does not exist as x approaches π/2: \(\lim_{x \rightarrow \frac{\pi}{2}} \left[\frac{\cos x}{\sin x}\right]\) does not exist.

Conclusion

Now that we know the limit of \(\left[\frac{\cos x}{\sin x}\right]\) as x approaches π/2 does not exist, it means that the limit of \(f(x)\) as x approaches π/2 does not exist. Therefore, Assertion (A) is false, and consequently, Reason (R) is irrelevant in this case.