Problem 126 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\lim _{x \rightarrow \infty} \frac{\sqrt{1-\cos 2 x}}{\sqrt{2} x}\) is (A) 1 (B) \(-1\) (C) Zero (D) Does not exist

Step-by-Step Solution

Answer

(C) Zero.

Step 1: Simplify the Trigonometric Expression

We start by rewriting the numerator \( \sqrt{1 - \cos 2x} \). The expression \( 1 - \cos 2x \) can be simplified using the identity \( 1 - \cos 2x = 2\sin^2 x \). Thus, \( \sqrt{1 - \cos 2x} = \sqrt{2\sin^2 x} = \sqrt{2} |\sin x| \).

Step 2: Rewrite the Limit

Substitute the simplified expression from Step 1 into the original limit. We have: \[ \lim _{x \rightarrow \infty} \frac{\sqrt{2} |\sin x|}{\sqrt{2} x} \]This simplifies to: \[ \lim _{x \rightarrow \infty} \frac{|\sin x|}{x} \]

Step 3: Evaluate the Limit

As \( x \) approaches infinity, the numerator \( |\sin x| \) oscillates between 0 and 1. However, the denominator \( x \) increases without bound. This means that the fraction \( \frac{|\sin x|}{x} \) approaches 0 because a bounded function divided by an unbounded function tends towards 0.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus for simplifying complex expressions. They are equations that are true for all values of the variables involved. In the given exercise, we utilized the identity \( 1 - \cos 2x = 2\sin^2 x \) to simplify the numerator \( \sqrt{1 - \cos 2x} \). This is one of many identities that help in transforming trigonometric expressions into alternatives that might be easier to work with. By recognizing and applying the appropriate identity, we can rewrite the expression as \( \sqrt{2} |\sin x| \), which sets the stage for further simplification.

Evaluating Limits

Limits are a fundamental concept in calculus that help us understand how a function behaves as it approaches a particular point or tends towards infinity. When evaluating limits, especially those involving infinity, it is crucial to analyze how both the numerator and denominator behave individually. In evaluating the limit \( \lim_{x \to \infty} \frac{|\sin x|}{x} \), for instance, we note that as \( x \) goes to infinity:

• The numerator \(|\sin x|\) oscillates between 0 and 1.
• The denominator \(x\) increases without bounds, moving towards infinity.

This pattern indicates that the fraction as a whole tends toward zero, as a bounded oscillating numerator over an unbounded denominator leads to a zero limit.

Infinite Limits

Infinite limits consider what happens as a function approaches infinity. In this exercise, \( x \) goes to infinity, which means we are analyzing the behavior as it "stretches" towards an extreme value. Infinite limits often require careful attention to how both parts of the function behave:

• If the numerator remains finite while the denominator grows infinitely large, the result is typically 0.
• Conversely, if the numerator grows without bound as well, further analysis may be needed to determine the limit's value.

Understanding the concept of infinity and how it affects limits is key to solving these types of problems. In our case, the behavior of \(|\sin x|\) limited the impact of the increasing \(x\), driving the overall expression towards zero.

Trigonometric Limits

Trigonometric limits are special cases where trigonometric functions play a key role. In the current limit problem, \(|\sin x|\) is the trigonometric component, which oscillates cyclically. Such behavior is common in trigonometric functions due to their periodic nature.To handle trigonometric limits, you often use established techniques:

• Identify and use trigonometric identities to simplify the expression.
• Evaluate the oscillation pattern, especially focusing on amplitude (range of values) and frequency.
• Determine the behavior as the whole expression approaches a given point or infinity.

By understanding and applying these principles, you can effectively analyze and solve limits involving trigonometric expressions, as we did to reach the conclusion that the given limit approaches 0.