Question

If $\mathrm{f}(\mathrm{x})=\lim _{\mathrm{n} \rightarrow \infty} \frac{2 \mathrm{x}}{\pi} \tan ^{-1} \mathrm{nx}\(, then value of \)\lim _{\mathrm{x} \rightarrow 0}[\mathrm{f}(\mathrm{x})-1]$ is, where [.] represents greatest integer function (A) 0 (B) \(-1\) (C) 1 (D) does not exist

Short answer

Question: Find the value of the limit \(\lim_{x \rightarrow 0} [f(x) - 1]\), where \(f(x) = \lim_{n \rightarrow \infty} \frac{2x}{\pi} \tan^{-1}(nx)\), and [.] represents the greatest integer function. Answer: -1

Step-by-step solution

Define f(x) using given limit

We are given that \(f(x) = \lim_{n \rightarrow \infty} \frac{2x}{\pi} \tan^{-1}(nx)\). Before proceeding, let's note that \(\tan^{-1}(nx)\) is the arctangent of \(nx\).

Analyze the limit

We need to find the limit \(\lim_{x \rightarrow 0} [f(x) - 1]\) as x approaches 0. We can rewrite this as: \(\lim_{x \rightarrow 0} [\lim_{n \rightarrow \infty} \frac{2x}{\pi} \tan^{-1}(nx) - 1]\).

Properties of arctangent

Recall that as \(y \rightarrow \infty\), \(\tan^{-1}(y) \rightarrow \frac{\pi}{2}\). This property will become useful in the next step, as we evaluate \(f(x)\) while \(n\) approaches infinity.

Evaluate f(x) while n approaches infinity

As \(n \rightarrow \infty\), we can rewrite \(f(x)\) as: $$f(x) = \lim_{n \rightarrow \infty} \frac{2x}{\pi} \tan^{-1}(nx) = \frac{2x}{\pi} \lim_{n \rightarrow \infty} \tan^{-1}(nx)$$ Now using the property of arctangent from Step 3, we get: $$f(x) = \frac{2x}{\pi} \cdot \frac{\pi}{2} = x$$

Evaluate the given limit using f(x)

Now we know \(f(x) = x\), let's substitute this back into the limit we need to find: $$\lim_{x \rightarrow 0} [f(x) - 1] = \lim_{x \rightarrow 0} [x - 1]$$

Applying the greatest integer function

As x approaches 0, \(x - 1\) approaches -1. Since the greatest integer function takes the largest integer less than or equal to the input, we have \([x - 1] = -1\) for \(x \rightarrow 0\). Therefore: $$\lim_{x \rightarrow 0} [x - 1] = -1$$

Conclusion

The limit \(\lim_{x \rightarrow 0} [f(x) - 1]\) is equal to -1. Therefore, the correct answer is (B) \(-1\).