Problem 119 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f(x)=\frac{1-\tan x}{4 x-\pi}, x \neq \frac{\pi}{4}, x \in\left[0, \frac{\pi}{2}\right] .\) If \(f(x)\) is continuous in \(\left[0, \frac{\pi}{2}\right]\), then \(f\left[\frac{\pi}{4}\right]\) is \(\quad\) [2004] (A) 1 (B) \(\frac{1}{2}\) (C) \(-\frac{1}{2}\) (D) \(-1\)

Step-by-Step Solution

Answer

Option (C) \(-\frac{1}{2}\).

Step 1: Understand the Exercise

We need to find the value of the function \( f(x) = \frac{1 - \tan x}{4x - \pi} \) at \( x = \frac{\pi}{4} \) given that the function is continuous on the interval \([0, \frac{\pi}{2}]\).

Step 2: Function Continuity Recall

For a function to be continuous at a point, it must satisfy the condition that the limit of the function as it approaches the point is equal to the function value at that point.

Step 3: Define the Concept of Continuity at \( x = \frac{\pi}{4} \)

We need to calculate the limit \( \lim_{x \to \frac{\pi}{4}} f(x) \) to define \( f\left(\frac{\pi}{4}\right) \) based on the given continuity. This limit calculation is necessary because the function is undefined directly at \( x = \frac{\pi}{4} \).

Step 4: Find \( \lim_{x \to \frac{\pi}{4}} f(x) \)

Substitute \( x = \frac{\pi}{4} \) into the function to check the indeterminate form: \( f(x) = \frac{1 - \tan x}{4x - \pi} \rightarrow \frac{1 - 1}{4(\frac{\pi}{4}) - \pi} = \frac{0}{0} \). This indicates we need to apply L'Hospital's Rule.

Step 5: Apply L'Hospital's Rule

Differentiate the numerator and denominator separately:- The derivative of the numerator \( 1 - \tan x \) is \(-\sec^2 x\).- The derivative of the denominator \( 4x - \pi \) is \(4\).Now calculate the limit: \[\lim_{x \to \frac{\pi}{4}} \frac{-\sec^2 x}{4} = \frac{-\sec^2 \frac{\pi}{4}}{4}\]

Step 6: Evaluate \( \sec^2 \frac{\pi}{4} \)

Compute \( \sec \frac{\pi}{4} = \frac{1}{\cos \frac{\pi}{4}} = \sqrt{2} \). Therefore, \( \sec^2 \frac{\pi}{4} = 2 \).

Step 7: Substitute and Simplify

Substitute back into the limit expression: \[\lim_{x \to \frac{\pi}{4}} f(x) = \frac{-2}{4} = -\frac{1}{2}\] Thus, by the continuity of \( f(x) \), \( f\left(\frac{\pi}{4}\right) = -\frac{1}{2} \).

Step 8: Select the Correct Option

The function \( f \) at \( x = \frac{\pi}{4} \) evaluates to \(-\frac{1}{2}\), which corresponds to option (C).

Key Concepts

Limits

In calculus, limits help us understand what happens to a function as its input approaches a certain value. This is crucial when dealing with points where the function is not directly defined. For the function \( f(x) = \frac{1 - \tan x}{4x - \pi} \), we are tasked to find the limit as \( x \to \frac{\pi}{4} \), because the function is not inherently defined at this point.

When we approach this specific value, and if the function tends to a single number, we call that number the limit. Transactions of this kind require understanding the behavior of both the numerator and the denominator of our rational function. If directly substituting the point into the function yields an indeterminate form like \( \frac{0}{0} \), further analysis with advanced tools such as L'Hospital's Rule is needed, making limits an integral part of working with continuous functions.

L'Hospital's Rule

L'Hospital's Rule is a powerful method in calculus that assists in finding limits of indeterminate forms, specifically \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). It capitalizes on taking derivatives of the numerator and denominator separately to resolve complexity in functions.

In our example, with \( f(x) = \frac{1 - \tan x}{4x - \pi} \), substituting \( x = \frac{\pi}{4} \) provided the indeterminate form \( \frac{0}{0} \). By differentiating the numerator \( 1 - \tan x \), we find \( -\sec^2 x \), and for the denominator \( 4x - \pi \), we get \( 4 \).

Applying L'Hospital's Rule:
\[\lim_{x \to \frac{\pi}{4}} \frac{-\sec^2 x}{4} \]

Next, by evaluating at \( x = \frac{\pi}{4} \), we can compute a definite limit.

Trigonometric Functions

Trigonometric functions form the very cornerstone of many calculus problems, linking angles to ratios of triangle sides. Specifically, tangent and secant functions appear frequently, each with unique properties and implications.

In the process of applying L'Hospital's Rule in the given exercise, knowing the derivatives and values of trigonometric functions proves essential.

For example, \( \tan x \) denotes the tangent of \( x \), and its derivative is \( \sec^2 x \), implementing growth characteristics of the tangent graph. Additionally, at \( x = \frac{\pi}{4} \), we know \( \tan\left(\frac{\pi}{4}\right) = 1 \).

The secant, which is \( \sec x = \frac{1}{\cos x} \), at \( \frac{\pi}{4} \), evaluates to \( \sqrt{2} \). Conclusively, \( \sec^2 \frac{\pi}{4} \) becomes \( 2 \), a crucial computation in calculating our limit with L'Hospital's Rule. Understanding these values is pivotal for problem-solving in continuous function analysis.