Problem 10 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

\(\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{\sqrt{2}-\sqrt{1+\cos x}}\) equals: \(\quad\) [April 8, 2019 (I)] (a) \(4 \sqrt{2}\) (b) \(\sqrt{2}\) (c) \(2 \sqrt{2}\) (d) 4

Step-by-Step Solution

Answer

The limit equals \(2 \sqrt{2}\) which corresponds to option (c).

Step 1: Identify the Limit to Evaluate

We are given the limit expression \( \lim_{x \rightarrow 0} \frac{\sin^2 x}{\sqrt{2} - \sqrt{1+\cos x}} \). Our task is to find the value of this limit as \( x \) approaches 0.

Step 2: Simplify the Expression

Starting with the denominator \( \sqrt{2} - \sqrt{1+\cos x} \), we rationalize it by multiplying the numerator and the denominator by the conjugate \( \sqrt{2} + \sqrt{1+\cos x} \).

Step 3: Rationalize the Denominator

Multiply \( \frac{\sin^2 x}{\sqrt{2} - \sqrt{1+\cos x}} \) by \( \frac{\sqrt{2} + \sqrt{1+\cos x}}{\sqrt{2} + \sqrt{1+\cos x}} \). The expression becomes \( \frac{\sin^2 x (\sqrt{2} + \sqrt{1+\cos x})}{2 - (1+\cos x)} \).

Step 4: Simplify the Fraction

The denominator simplifies to \( 1 - \cos x \), and \( 1 - \cos x \) can be expressed as \( 2\sin^2(\frac{x}{2}) \) using the trigonometric identity.

Step 5: Substitute Trigonometric Identity

The whole expression is \( \frac{\sin^2 x (\sqrt{2} + \sqrt{1+\cos x})}{2\sin^2(\frac{x}{2})} \). Now, substitute \( \sin^2 x = (2\sin(\frac{x}{2})\cos(\frac{x}{2}))^2 \).

Step 6: Cancel out Common Terms

After substitution, calculate and simplify the terms to cancel out \( \sin(\frac{x}{2}) \) between numerator and denominator. It reduces the expression to \( \frac{(2\cos(\frac{x}{2}))^2(\sqrt{2} + \sqrt{1+\cos x})}{2} \).

Step 7: Evaluate the Limit

As \( x \rightarrow 0 \), \( \cos(\frac{x}{2}) \rightarrow 1 \) and \( \sqrt{1+\cos x} \rightarrow \sqrt{2} \). So the limit evaluates to \( \frac{4 \sqrt{2}}{2} = 2\sqrt{2} \).

Key Concepts

Trigonometric Limits

Understanding trigonometric limits is crucial in calculus, particularly when dealing with functions involving sine, cosine, tangents, etc. Limits help us determine the behavior of functions as they approach a given point. In this exercise, the limit is evaluated as \( x \rightarrow 0 \).
When dealing with trigonometric functions, it’s important to remember some key limits. For instance:

• \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \)
• \( \lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0 \)

In this problem, we’re primarily working with \( \sin^2(x) \) and \( \cos(x) \). As \( x \) approaches zero, we need to manipulate the function to reveal forms that allow us to apply these limits effectively. Keeping these standard limits in mind helps us simplify complex trigonometric expressions, and is especially useful in rationalizing denominators or evaluating limits under tricky conditions.

Rationalizing Techniques

Rationalizing is a technique used to eliminate roots from the denominator. It simplifies the function and makes finding limits more manageable.
In this exercise, the denominator \( \sqrt{2} - \sqrt{1+\cos x} \) can be tricky due to the root. To handle this, we use the conjugate \( \sqrt{2} + \sqrt{1+\cos x} \). Multiplying the original expression by this conjugate over itself changes our expression to: \[ \frac{\sin^2 x (\sqrt{2} + \sqrt{1+\cos x})}{(\sqrt{2} - \sqrt{1+\cos x})(\sqrt{2} + \sqrt{1+\cos x})}\]
By doing this, the denominator simplifies to a difference of squares: \( 2 - (1+\cos x) = 1 - \cos x \). This is a common strategy and often paves the way for further simplification using identities or transformations.

Trigonometric Identities

Trigonometric identities are invaluable tools in simplifying expressions and solving calculus problems. They allow us to transform tricky trigonometric expressions into simpler forms that are more manageable.
In the problem at hand, we recognize that the expression \( 1 - \cos x \) can be rewritten using the identity:\[1 - \cos x = 2\sin^2\left(\frac{x}{2}\right)\] This identity is crucial because it allows us to cancel terms and simplify the original expression. Additionally, the identity \( \sin^2 x = (2\sin(\frac{x}{2})\cos(\frac{x}{2}))^2 \) helps further simplify our expression after substituting into the equation: \[\frac{\sin^2 x (\sqrt{2} + \sqrt{1+\cos x})}{2\sin^2\left(\frac{x}{2}\right)}\] By substituting and simplifying, trigonometric identities reduce the complexity of the problem, making it easier to compute or evaluate as \( x \to 0 \). Recognizing and applying the right identity is a key skill when working with trigonometric limits.