Problem 37 - A Complete Resource Book in Mathematics for JEE Main

Question

If the function \(f(x)=\left\\{\begin{array}{cc}(1-|\tan x|)^{\frac{a}{\tan x \mid}} & \frac{-\pi}{4}

Step-by-Step Solution

Answer

(D) None of these

Step 1: Identify Continuity Condition

For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the value of the function at that point must be equal. We will check these for continuity at \(x = 0\).

Step 2: Calculate Left-Hand Limit as x Approaches 0

We examine \(f(x)\) as \(x\) approaches 0 from the left, which implies \(-\frac{\pi}{4} < x < 0\). The relevant expression is \((1 - |\tan x|)^{\frac{a}{|\tan x|}}\). As \(x \to 0^-\), \(\tan x \to 0\), so \(1 - |\tan x| \to 1\) and \(\frac{a}{|\tan x|} \to \infty\). Evaluating this limit requires observing that \(\lim_{{x \to 0^-}} (1 - |\tan x|)^\infty = 0\) if \(a < 0\). In this problem, setting \(\lim_{{x \to 0^-}} f(x) = 0 \) can help continue.

Step 3: Calculate Right-Hand Limit as x Approaches 0

We examine \(f(x)\) as \(x\) approaches 0 from the right, \(0 < x < \frac{\pi}{4}\). The relevant expression is \(\frac{\sin 3x}{e^{\sin 2x}}\). As \(x \to 0^+\), both \(\sin 3x\) and \(e^{\sin 2x}\) approach 0 and 1, respectively, so \(\lim_{{x \to 0^+}} \frac{\sin 3x}{e^{\sin 2x}} = \frac{0}{1} = 0\).

Step 4: Evaluate Function at x equals 0

At \(x = 0\), the function \(f(x) = b\). Hence, the condition for continuity at \(x = 0\) is \(\lim_{{x \to 0^-}} f(x) = \lim_{{x \to 0^+}} f(x) = f(0) = b = 0\).

Step 5: Assess Values of a and b from Options

Since all options for \(b\) are non-zero from (A), (B), and (C), only an option with \(b=0\) can be correct from continuity conditions. But comparing, we see no options have \(b = 0\). Therefore, none of the given options satisfy all continuity conditions.

Key Concepts

Understanding Limits of Trigonometric Functions

When working with limits involving trigonometric functions, it is essential to understand the behavior of these functions as the variable approaches a particular point. Let's take for example, the case where the function involves \( \tan x \), which behaves uniquely around specific points.

• As \, \( x \to 0 \), the behavior of \, \( \tan x \to 0 \) \, is crucial in determining the expression's overall limit. This is because \, \( \tan x \) \, approaches zero, implying that any function depending on this limit may also approach very small values or diverge to infinity.
• Furthermore, examining a function like \, \( \frac{\sin 3x}{e^{\sin 2x}}\) \, as \( x \to 0 \) \, shows how both the numerator and denominator play a role. With \, \( \sin 3x \to 0 \) \, and \, \( e^{\sin 2x} \approx 1 \), the limit approaches zero.

It’s important to practice calculating these limits to gain intuition about how they contribute to the continuity of functions focusing on regions around special angles.

Decoding Piecewise Functions for Continuity

Piecewise functions are defined differently across their domain based on set intervals and can sometimes be challenging when assessing continuity. Each piece provides a unique expression for the function over different portions of the domain.
In considering the function \, \( f(x)\) \, provided:

• For \, \( -\frac{\pi}{4} < x < 0 \), the function is defined as \, \( (1 - |\tan x|)^{\frac{a}{|\tan x|}} \).
• At \, \( x = 0 \), the function has a set value, \, \( f(x) = b \).
• And for \, \( 0 < x < \frac{\pi}{4} \), \, \( \frac{\sin 3x}{e^{\sin 2x}} \).

To ensure the entire function is continuous at a boundary point like \, \( x = 0 \), the left and right limits should match the value of the function at that point. This is essential for recognizing how different pieces must 'fit' together smoothly.

Mastering Evaluation of Limits

Evaluating limits is a fundamental process to determine the behavior of a function at points where it could potentially be undefined or exhibit unusual behavior. The step-by-step limit evaluation for continuity can involve several crucial calculations and observations.
First, identify the specific type of limit you're working with:

• Left-hand limits evaluate the tendency of the function as it approaches a point from the left.
• Right-hand limits cover the function's behavior from the right side of a point.
• Function value directly provides the output of the function, if defined at the point in question.

For a continuous function, these three values must be equal at the point of interest. Thus, each limit and function value must be evaluated meticulously to ensure they meet continuity criteria, particularly at juxtaposing boundaries with different expressions—like in a piecewise function.