Problem 3 - A Textbook of Differential Calculus

Question

If \(I_{1}=\lim _{x \rightarrow \infty}\left(\tan ^{-1} \pi x-\tan ^{-1} x\right) \cos x\) and \(I_{2}=\lim _{x \rightarrow 0}\left(\tan ^{-1} \pi x-\tan ^{-1} x\right) \cos x\), then \(\left(I_{1}, I_{2}\right)\) is (a) \((0,0)\) (b) \((0,1)\) (c) \((1,0)\) (d) None of these

Step-by-Step Solution

Answer

(0,0) is the solution for \((I_1, I_2)\).

Step 1: Define the Limits

First, we should identify the expressions for which we need to find the limits:\( I_1 = \lim_{x \rightarrow \infty}(\tan^{-1}(\pi x) - \tan^{-1}(x))\cos(x) \) and \( I_2 = \lim_{x \rightarrow 0}(\tan^{-1}(\pi x) - \tan^{-1}(x))\cos(x) \).

Step 2: Evaluate Limit for I_1 as x approaches infinity

Consider \( I_1 = \lim_{x \rightarrow \infty}(\tan^{-1}(\pi x) - \tan^{-1}(x))\cos(x) \). As \( x \to \infty \), \( \tan^{-1}(\pi x) \to \frac{\pi}{2} \) and \( \tan^{-1}(x) \to \frac{\pi}{2} \) as well. Thus, \( \tan^{-1}(\pi x) - \tan^{-1}(x) \to 0 \). Also, \( \cos(x) \) oscillates between -1 and 1. Therefore, \( I_1 = \lim_{x \rightarrow \infty} 0\cdot \cos(x) = 0 \).

Step 3: Evaluate Limit for I_2 as x approaches zero

Next, consider \( I_2 = \lim_{x \rightarrow 0}(\tan^{-1}(\pi x) - \tan^{-1}(x))\cos(x) \). As \( x \to 0 \), \( \tan^{-1}(\pi x) \approx \pi x \) and \( \tan^{-1}(x) \approx x \). So, \( \tan^{-1}(\pi x) - \tan^{-1}(x) \approx \pi x - x = (\pi - 1)x \). The limit becomes \( I_2 = \lim_{x \rightarrow 0}((\pi - 1)x)\cos(x) \approx \lim_{x \rightarrow 0}((\pi - 1)x) \approx 0 \). Since \( \cos(x) \approx 1 \) when \( x \to 0 \), the term remains negligible, giving \( I_2 = 0 \).

Step 4: Summarize the Results

From the evaluations, we have \( I_1 = 0 \) and \( I_2 = 0 \). Therefore, the pair \((I_1, I_2) = (0, 0)\). Compare this result with the given options to find the correct answer.

Key Concepts

Limits

Limits are a fundamental concept in differential calculus. They help us understand the behavior of functions as they approach a specific point. When we evaluate the limit of a function as the variable approaches infinity or zero, we're essentially examining its trend or end behavior at extreme values.

When the limit of a function is zero, it indicates that as the input gets very large or very small, the output of the function nears zero. Evaluating limits, such as in this exercise, involves observing the behavior of each component of the expression to determine its resulting value.

Certain functions can go towards infinity, zero or a constant, and recognizing these patterns is key to understanding limits. The concept of limits doesn't just help in evaluating simple functions, but complex expressions as well, making it a cornerstone of calculus.

Inverse Trigonometric Functions

Inverse trigonometric functions, like \(\tan^{-1}\), play a crucial role in various mathematical and engineering fields. These functions help in determining angles when given a trigonometric value.

1. **Range and Principal Values**: Understanding the range of inverse trigonometric functions is essential. For instance, \(\tan^{-1}(x)\) ranges between \(-\pi/2\) and \(\pi/2\), implying that for any real \(x\), \(\tan^{-1}(x)\) will always yield a result within this interval.

2. **Behavior as \(x\) Changes**: These functions have specific behaviors as \(x\) approaches certain values: - As \(x\) approaches infinity, \(\tan^{-1}(x)\) approaches \(\pi/2\). - As \(x\) approaches zero, \(\tan^{-1}(x)\) approaches zero too.

Understanding these properties aids in solving problems involving inverse trigonometric expressions, like those seen in limit evaluations.

Evaluation of Limits

Evaluating limits that involve trigonometric expressions often requires recognizing how the components of the limit behave as \(x\) approaches a particular value.

- **Matching Terms**: When evaluating limits, especially when inverse trigonometric functions are involved, it’s crucial to observe how linear terms interact. As in the example of \(\tan^{-1}(\pi x) - \tan^{-1}(x)\), subtraction leads to simplification as common behaviors at extreme \(x\) values get highlighted.

- **Simplification Tactics**: Sometimes, it becomes necessary to approximate complex expressions, especially as \(x\) approaches zero or infinity. In the evaluated problem, understanding that \(\tan^{-1}(\pi x) - \tan^{-1}(x) \approximates (\pi - 1)x\) is essential for using standard limit laws effectively.

Using these strategies enables a clear understanding of how limits behave, ensuring accuracy when evaluating complex calculus problems.