Problem 38 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

\(\lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{1 / x}\) is equal to: \(\quad\) [Sep. \(\mathbf{0 2}, \mathbf{2 0 2 0}\) (II)] (a) \(e\) (b) 2 (c) 1 (d) \(e^{2}\)

Step-by-Step Solution

Answer

The limit is \( e^2 \), so the answer is (d).

Step 1: Understanding the Problem

We need to find \( \lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{1 / x} \). This is a limit problem involving an exponential limit type, which generally points towards the use of the limit definition of the exponential function or logarithms.

Step 2: Simplifying the Expression

Notice that when \( x = 0 \), \( \tan \left(\frac{\pi}{4} + x\right) = \tan \left(\frac{\pi}{4}\right) = 1 \). However, since \( \lim _{x \rightarrow 0} 1^{1/x} \) can be potentially indeterminate, we need another approach.

Step 3: Logarithmic Transformation

Take the natural log of the expression to simplify: \[ \ln\left( \left(\tan\left(\frac{\pi}{4} + x\right)\right)^{1/x} \right) = \frac{\ln\left( \tan\left(\frac{\pi}{4} + x\right)\right)}{x}. \] This is the form \( \frac{0}{0} \) as \( x \to 0 \), a classic case for applying L'Hôpital's Rule.

Step 4: Applying L'Hôpital's Rule

Using L'Hôpital's Rule, differentiate the numerator and denominator: - Derivative of numerator \( \ln\left( \tan\left(\frac{\pi}{4} + x\right)\right)\) is \( \frac{1}{\tan\left(\frac{\pi}{4} + x\right)} \cdot \sec^2\left(\frac{\pi}{4} + x\right) \). - Derivative of denominator \( x \) is \( 1 \).Then the limit becomes \[ \lim_{x \to 0} \frac{\sec^2\left(\frac{\pi}{4} + x\right)}{\tan\left(\frac{\pi}{4} + x\right)}. \]

Step 5: Evaluating at the Limit

As \( x \to 0 \), \( \tan\left(\frac{\pi}{4} + x\right) \to 1 \) and \( \sec^2\left(\frac{\pi}{4} + x\right) \to 2 \). Thus, the limit simplifies to: \[ \lim_{x \to 0} \frac{2}{1} = 2. \] This means the original natural logarithm limit is \( 2 \).

Step 6: Converting Back from Logarithmic Form

Since we initially transformed the expression using a logarithm, we have \( \ln(L) = 2 \) where \( L \) is the original limit. Therefore, exponentiate to find \( L = e^2 \).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for evaluating limits that yield an indeterminate form. Usually, these forms are either \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When you encounter such forms, L'Hôpital's Rule allows you to differentiate the numerator and the denominator separately and then find the limit again. Thus,

• If \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0} \), then \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \), provided the right-hand side limit exists.
• This rule is very useful when faced with complex functions that are hard to simplify directly.

For example, in the problem, after taking the natural logarithm of the function, we obtain a \( \frac{0}{0} \) form. By applying L'Hôpital's Rule, we can differentiate both the numerator and denominator to get a form that is easier to evaluate and solve.

Exponential Limits

Exponential limits involve expressions that grow exponentially or decay exponentially. They often appear in the format of raised powers involving variables, such as \( x^{1/x} \) or \( (some extstyle{ }function)^{1/x} \). To evaluate limits of this type, one common technique is to use logarithms to simplify the expression:

• Take the natural logarithm of the entire expression, simplifying its complexity.
• Transform the exponent into a product, which simplifies differentiation.

This approach converts the exponential expression into a form where calculus tools like L'Hôpital's Rule can be applied. In our case, taking the logarithm changes \( \ln((\tan(\frac{\pi}{4} + x))^{1/x}) \) into a simpler fraction. Once evaluated, the result is exponentiated to convert back from the logarithm form, ultimately giving us the solution \( e^2 \).

Trigonometric Limits

Trigonometric limits often involve functions like sine, cosine, or tangent. These functions can behave unpredictably around certain points, especially near zero or singularities. When tackled within limit problems:

• The key is to remember the fundamental properties and identities, such as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and small-angle approximations like \( \sin(x) \approx x \) for small \( x \).
• Considering transformations, like \( \tan(\frac{\pi}{4} + x) \) in our exercise, which is affected by the changes made to angular values, helps predict behavior as \( x \to 0 \).

In the original problem, understanding how \( \tan \) behaves when approaching certain angles, particularly how \( \tan(\frac{\pi}{4} + x) \to 1 \) as \( x \to 0 \), allows us to evaluate the limits accurately through derivative differentiation and simplification.