Problem 72 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\lim _{x \rightarrow 0} \frac{\sqrt{1-\sqrt{\cos x}}}{x}=\) (A) \(\frac{1}{2}\) (B) \(-\frac{1}{2}\) (C) Does not exist (D) None of these

Step-by-Step Solution

Answer

The answer is (A) \( \frac{1}{2} \).

Step 1: Understand the Limits Problem

You need to find the limit of the function \( \frac{\sqrt{1-\sqrt{\cos x}}}{x} \) as \( x \to 0 \). This involves determining the behavior of the expression as \( x \) becomes very small.

Step 2: Simplify the Expression

Look at the expression \( \sqrt{1-\sqrt{\cos x}} \). When \( x \to 0 \), \( \cos x \approx 1 - \frac{x^2}{2} \) using the first term of the Taylor series expansion for cosine.

Step 3: Apply Taylor Series Approximation

Substitute the approximation into the expression: \( \sqrt{\cos x} \approx \sqrt{1 - \frac{x^2}{2}} \approx 1 - \frac{x^2}{4} \) using binomial expansion.

Step 4: Simplify Further

Rewrite \( 1 - \sqrt{\cos x} \approx \frac{x^2}{4} \). Then \( \sqrt{1-\sqrt{\cos x}} \approx \sqrt{\frac{x^2}{4}} \).

Step 5: Solve the Limit

Simplifying \( \sqrt{\frac{x^2}{4}} = \frac{|x|}{2} \). As \( x \to 0 \), the absolute value \( |x| \to x \), thus \( \frac{|x|}{2x} \to \frac{1}{2} \).

Step 6: Evaluate the Final Expression

The limit \( \lim_{x \to 0} \frac{\sqrt{1 - \sqrt{\cos x}}}{x} = \frac{1}{2} \).

Key Concepts

Taylor series

Taylor series is an essential concept in calculus that allows us to approximate more complex functions with polynomials. This is especially helpful when evaluating limits, derivatives, or integrals where direct computation might be challenging.
The basic idea is to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For example, the Taylor series expansion of the cosine function around zero (called a Maclaurin series) is given by:

• \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \ldots \)

In our problem, we use the first term, \( \cos x \approx 1 - \frac{x^2}{2} \), since it's enough to capture the behavior of \( \cos x \) as \( x \to 0 \). This simplification is particularly useful for our limit evaluation.

Binomial expansion

The binomial expansion is a way to expand expressions raised to a power. It’s especially handy for approximating expressions like \( (1 + u)^n \) when |u| is small.
The binomial expansion is given by:

• \( (1 + u)^n = 1 + nu + \frac{n(n-1)}{2!}u^2 + \ldots \)

For small values of \( u \), we often only need the first few terms. In the step-by-step solution, we applied binomial expansion to \( \sqrt{1 - \frac{x^2}{2}} \) to simplify it to \( 1 - \frac{x^2}{4}\), since the multiplier for the first term is minimal, making the approximation easier and more accurate as \( x \) approaches zero.

Trigonometric limits

Trigonometric limits often involve the behavior of sine and cosine functions as the variable approaches zero. These limits are crucial because many other functions can be expressed in terms of trigonometric ones.
In the case of our exercise, understanding the limit behavior of \( \cos x \) as \( x \) approaches 0 is essential. We used the known approximation \( \cos x \approx 1 - \frac{x^2}{2} \) to rewrite the nested function \( \sqrt{1-\sqrt{\cos x}}\).
Such trigonometric limits might initially appear complex, but they simplify dramatically when recognized as standard limit forms or through relevant approximations like Taylor series.

Indeterminate forms

Indeterminate forms are expressions where substitution directly into the limit does not yield a clear answer. The form \( \frac{0}{0} \) is one of the most common types in calculus.
In the given problem, direct substitution of \( x = 0 \) into \( \frac{\sqrt{1-\sqrt{\cos x}}}{x} \) gives \( \frac{0}{0} \), indicating an indeterminate form. This signals the need for further analysis or simplification to evaluate the limit correctly.
To resolve the indeterminate form in our exercise, various approximation techniques are used, such as Taylor series and binomial expansion, to redefine the expression in terms of \( x \).
Understanding indeterminate forms is crucial because it informs us when simple plugging in of limit values isn't enough, prompting us to use mathematical tools and techniques to understand the behavior near the limit point fully.