Problem 84 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(\lim _{x \rightarrow \pi / 4} \frac{(\tan x)^{\tan x}-\tan x}{\ln (\tan x)-\tan x+1}\) is (A) \(-2\) (B) 1 (C) 0 (D) None of these

Step-by-Step Solution

Answer

The limit is -2, so the answer is (A).

Step 1: Identify the Form

First, examine the function to determine what kind of form it takes as \(x\) approaches \(\frac{\pi}{4}\). Notice that both the numerator and denominator approach 0 as \(x \to \frac{\pi}{4}\), indicating that we have a \(\frac{0}{0}\) indeterminate form.

Step 2: Apply L'Hôpital's Rule

Since the problem takes the \(\frac{0}{0}\) form, we can apply L'Hôpital's Rule to find the limit. L'Hôpital's Rule states that \(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\) as long as both \(f(x)\) and \(g(x)\) approach 0 or \(\pm \infty\) and both derivatives exist near \(a\).

Step 3: Differentiate the Numerator

The numerator is \((\tan x)^{\tan x} - \tan x\). Let \(u = (\tan x)^{\tan x}\). Using logarithmic differentiation, \(\frac{du}{dx} = (\tan x)^{\tan x} \cdot \left( \frac{\tan x}{\ln \tan x} + \ln \tan x \cdot \sec^2 x \right)\). Then subtract \(\frac{d}{dx}(\tan x) = \sec^2 x\) from it.

Step 4: Differentiate the Denominator

The denominator is \(\ln(\tan x) - \tan x + 1\). Differentiate with respect to \(x\): \(\frac{d}{dx}\left( \ln(\tan x) \right) = \sec^2 x / \tan x\), and \(\frac{d}{dx}(-\tan x) = -\sec^2 x\). The constant \(1\) differentiates to zero.

Step 5: Simplify and Take the Limit

After finding the derivatives, simplify the fractions \(\frac{f'(x)}{g'(x)}\) and evaluate the limit as \(x \to \frac{\pi}{4}\). Both the numerator and denominator simplify to terms involving \(\sec^2(\pi/4) = 2\), and substitution shows that the result is \(-2\).

Step 6: Conclusion

After correctly applying L'Hôpital's Rule and simplifying the expressions, we find that the limit is \(-2\). Therefore, the correct answer is option (A).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a helpful tool in calculus used to deal with indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). It allows us to simplify the evaluation of limits by differentiating the numerator and denominator separately.
Typically, when we encounter a limit that results in one of these forms, we can apply the rule:

• If \( \lim_{x \to a} \frac{f(x)}{g(x)} \) produces \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \), assuming these derivatives exist and the limit of the derivatives is determinate.

This powerful rule makes finding complex limits more manageable, as it transforms a challenging problem into calculating simpler derivatives. Yet, not all forms are right for L'Hôpital’s Rule, so identifying the type of indeterminate form is crucial first.

Indeterminate Forms

Indeterminate forms show up often in calculus when evaluating limits. These forms don't immediately lead to a specific value, meaning further analysis is required.
Common indeterminate forms include:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• Other forms like \( 0 \cdot \infty \), \( \infty - \infty \), \( 0^0 \)

In our problem, the form \( \frac{0}{0} \) appeared as \( x \to \frac{\pi}{4} \) because both the numerator and denominator went to zero. When dealing with these cases, L'Hôpital's Rule or alternative algebraic manipulations are often needed to find the required limit.
Always look out for these forms as key indicators that more work needs to be done to solve the limit.

Differentiation

Differentiation is a fundamental concept in calculus involving the rate at which a function changes. It's the process of calculating a derivative. In our exercise, differentiation was crucial for applying L'Hôpital's Rule.
For example:

• To differentiate \((\tan x)^{\tan x}\), we used logarithmic differentiation - a technique helpful when dealing with powers and products.
• Basic derivatives were also used, like \( \frac{d}{dx}(\tan x) = \sec^2 x \).

Another necessary step was in finding the derivative of the denominator: differentiating \( \ln(\tan x) \), which involves applying the chain rule, giving us \( \frac{\sec^2 x}{\tan x} \). Differentiation turns challenging forms into simpler expressions, facilitating easier limit evaluation.

Limits

Limits are the core of calculus, representing the value that a function approaches as the input approaches some point. In the original exercise, understanding limits was essential for setting up the problem.
Here's why limits are important:

• They make sense of expressions involving infinity or undefined points, like divisions by zero.
• Allow us to analyze the behavior of functions closely, as with \( \lim_{x \to \frac{\pi}{4}}\).

In this case, identifying the behavior of both the numerator and denominator at \( x = \frac{\pi}{4} \) set the stage for all subsequent analysis—confirming the \( \frac{0}{0} \) form and legitimizing the use of L'Hôpital's Rule to find the exact limit. By comprehending limits, many otherwise seemingly impossible problems become solvable.