Problem 103 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The function \(f(x)=\frac{1-u^{2}}{2+u^{2}}\), where \(u=\tan x\), has (A) discontinuity of the first kind at \(x=n \pi \pm \frac{\pi}{2}\), \(n \in I\) (B) discontinuity of the second kind at \(x=n \pi \pm \frac{\pi}{2}\), \(n \in I\) (C) removable discontinuity at \(x=n \pi \pm \frac{\pi}{2}, n \in I\) (D) continuous at \(x=n \pi \pm \frac{\pi}{2}, n \in I\)

Step-by-Step Solution

Answer

The function has discontinuity of the second kind at \( x=n\pi \pm \frac{\pi}{2} \).

Step 1: Understanding the Function

We have the function \( f(x) = \frac{1-u^2}{2+u^2} \), where \( u = \tan x \). This means \( f(x) = \frac{1-\tan^2 x}{2+\tan^2 x} \). We need to investigate the behavior of this function at certain points.

Step 2: Identifying Critical Points

The critical points we need to check are \( x = n\pi \pm \frac{\pi}{2} \), where \( n \in I \). At these points, \( \tan x \) has vertical asymptotes, implying \( u = \tan x \) goes to infinity.

Step 3: Analyzing Discontinuity at Critical Points

Since \( \tan x \) tends to infinity at \( x = n\pi \pm \frac{\pi}{2} \), the denominator \( 2+\tan^2 x \) becomes very large. However, for such discontinuity, consider evaluating limits from both sides of these points.

Step 4: Evaluating One-Sided Limits

Evaluate the one-sided limits as \( x \to n\pi + \frac{\pi}{2}^- \) and \( x \to n\pi + \frac{\pi}{2}^+ \). As \( x \) approaches these points, the \( u^2 \) term in both the numerator and denominator becomes very large, leading both terms potentially to infinity.

Step 5: Determining Limit Divergence

As the limits could diverge from different sides or yield non-matching values, this mismatch indicates discontinuity of the second kind due to the jump and potential infinity differences at each critical point.

Key Concepts

Discontinuity of the Second Kind

Discontinuity of the second kind, also known as an essential or infinite discontinuity, occurs when the limits from the left and right of a certain point do not exist or are not equal. Within the context of the function \( f(x) = \frac{1-\tan^2 x}{2+\tan^2 x} \), we discover such discontinuities at the critical points \( x = n\pi \pm \frac{\pi}{2} \), where \( n \) is an integer.

Here's how to identify it:

• These critical points represent values where \( \tan x \) exhibits vertical asymptotes.
• At these points, \( \tan x \to \infty \) or \( \tan x \to -\infty \).
• Given these behaviors, both the numerator \( 1-\tan^2 x \) and the denominator \( 2+\tan^2 x \) can simultaneously trend toward infinity in different ways.

The discrepancy in limits from the left and right sides at these points confirms a discontinuity of the second kind, indicating that the function cannot be smoothly connected at these junctures.

Behavior of Trigonometric Functions

Trigonometric functions, such as the tangent function in our case, often exhibit intriguing behaviors around special angles. Understanding these patterns is crucial for identifying discontinuities in functions that depend on trigonometric expressions.

When analyzing \( \tan x \), recall:

• \( \tan x \) has vertical asymptotes at every odd multiple of \( \frac{\pi}{2} \), expressed as \( x = n\pi + \frac{\pi}{2}, n \in \mathbb{Z} \).
• As \( x \to \frac{\pi}{2} \) from the left (\( x \to \frac{\pi}{2}^- \)), \( \tan x \to +\infty \).
• Conversely, for \( x \to \frac{\pi}{2}^+ \), \( \tan x \to -\infty \).

These opposing directions cause the trigonometric function's behavior to diverge dramatically around these points, which can lead to discontinuities in related functions.

Limits and Asymptotic Behavior

Understanding how limits work is essential in analyzing the continuity or discontinuity of functions. When dealing with trigonometric functions like tangent, limits can reveal much about asymptotic behavior.

As \( x \to n\pi \pm \frac{\pi}{2} \) for \( n \in \mathbb{Z} \), watch how the limits behave:

• If \( \tan x \to \infty \), both the numerator \( 1-\tan^2 x \) and the denominator \( 2+\tan^2 x \) may diverge differently or not align.
• The differing tendencies for \( x \to n\pi + \frac{\pi}{2}^- \) and \( x \to n\pi + \frac{\pi}{2}^+ \), where the function approaches different infinities, is key.

When limits from both approaches do not match or don't exist, this signals discontinuity. Analyzing these divergent trends demonstrates how limits and asymptotic behaviors underline the presence of essential discontinuities.