Question

The value of \[ \lim_{x \to \infty} \left( \frac{\pi}{2} - \tan^{-1} x \right)^{1/x} \] is:

• A. 0
• B. 1
• C. -1
• D. e

Hint:

When evaluating limits of the form \( \left( \text{expression} \right)^{1/x} \), if the base approaches zero, we can approximate it and simplify the expression to find the limit.

Answer 1

The Correct Option is B

We are tasked with finding the value of the limit: \[ \lim_{x \to \infty} \left( \frac{\pi}{2} - \tan^{-1} x \right)^{1/x}. \] Step 1: Behavior of \( \tan^{-1} x \) as \( x \to \infty \). We know that as \( x \to \infty \), the arctangent function behaves as: \[ \tan^{-1} x \to \frac{\pi}{2}. \] So, the expression \( \frac{\pi}{2} - \tan^{-1} x \) approaches zero as \( x \to \infty \). Step 2: Substituting into the limit. Substitute \( \frac{\pi}{2} - \tan^{-1} x \) into the limit expression: \[ \lim_{x \to \infty} \left( \frac{\pi}{2} - \tan^{-1} x \right)^{1/x}. \] We now need to examine the behavior of the quantity \( \left( \frac{\pi}{2} - \tan^{-1} x \right)^{1/x} \) as \( x \to \infty \). Since \( \frac{\pi}{2} - \tan^{-1} x \) approaches 0, we are dealing with the limit of a form similar to \( 0^{1/x} \). Step 3: Simplifying the expression. For very large \( x \), \( \frac{\pi}{2} - \tan^{-1} x \) can be approximated by the following expansion: \[ \frac{\pi}{2} - \tan^{-1} x \approx \frac{1}{x}. \] Thus, the expression becomes: \[ \lim_{x \to \infty} \left( \frac{1}{x} \right)^{1/x}. \] This simplifies to: \[ \lim_{x \to \infty} x^{-1/x}. \] We know that \( x^{-1/x} \) approaches 1 as \( x \to \infty \).
Step 4: Conclusion. Therefore, the value of the limit is 1, and the correct answer is (b).