Problem 29 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

\(\lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos \\{2(x-2)\\}}}{x-2}\right)\) (a) equals \(\sqrt{2}\) (b) equals \(-\sqrt{2}\) (c) equals \(\frac{1}{\sqrt{2}}\) (d) does not exist

Step-by-Step Solution

Answer

The limit equals \(\sqrt{2}\).

Step 1: Recognize Indeterminate Form

When evaluating the limit \( \lim_{x \to 2} \frac{\sqrt{1-\cos{2(x-2)}}}{x-2} \), substituting \(x = 2\) directly into the expression gives \(\frac{\sqrt{1-\cos{0}}}{0} = \frac{0}{0}\), which is an indeterminate form. Therefore, we need to simplify the expression to evaluate the limit.

Step 2: Trigonometric Identity

We use the identity \( 1 - \cos{\theta} = 2\sin^2{\frac{\theta}{2}} \). Apply this to the expression: \( 1 - \cos{2(x-2)} = 2\sin^2{(x-2)} \). The expression simplifies to \( \frac{\sqrt{2\sin^2(x-2)}}{x-2} \).

Step 3: Simplification Under the Square Root

Simplify \( \sqrt{2\sin^2(x-2)} \) to \( \sqrt{2}|\sin(x-2)| \). The expression becomes \( \frac{\sqrt{2}|\sin(x-2)|}{x-2} \). Since \(x\) approaches 2 from either direction, \(\sin(x-2)\) is approximately \(x-2\), we remove the absolute value, giving \( \frac{\sqrt{2}\sin(x-2)}{x-2} \).

Step 4: Resolve the Indeterminate Limit

We recognize \( \frac{\sin(x-2)}{x-2} \) as a form of \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1\). Therefore, the expression simplifies to \( \sqrt{2} \cdot 1 = \sqrt{2} \) when taking the limit as \( x \to 2 \).

Key Concepts

Trigonometric Limits

Exploring limits involving trigonometric functions can initially seem daunting. When faced with these types of problems, the values are often paired with trigonometric expressions within a limit function.

The objective of finding the limit of a trigonometric function, like in the problem \( ext{lim}_{x \to 2} \left(\frac{\sqrt{1-\cos{2(x-2)}}}{x-2}\right)\), is to determine what the expression approaches as \(x\) nears the specified value. In this case, it's as \(x\) approaches 2.

• First task: Recognize the type of trigonometric expression being handled.
• Second task: Simplify the trigonometric function, often using trigonometric identities, to ensure a smooth evaluation.

By doing this, unbeknownst hurdles created by trigonometry within a limit can be better managed, providing a unique and instructive approach to evaluating limits.

Indeterminate Forms

In calculus, finding that an expression simplifies into an indeterminate form like \(\frac{0}{0}\) indicates that direct substitution does not yield a solution. Indeterminate forms necessitate deeper examination and further algebraic manipulation to find the limit.

For instance, in the given problem, substituting \(x = 2\) results in \(\frac{0}{0}\). This tells us that we cannot immediately determine the limit's value. Instead, by recognizing the presence of an indeterminate form, we are prompted to use algebraic steps or trigonometric identities to rewrite the function in a way that can be computed.

• Indeterminate forms are signals. When calculations yield \(0/0\), \(\infty/\infty\), or other types, it suggests potential simplifications.
• Achieving successful simplification requires recognizing the opportunities to reformat to familiar functions, like \(\frac{\sin{x}}{x} = 1\), which can be resolved through limits.

This procedure is often accompanied by techniques such as l'Hôpital's Rule or using identities to transform the scenario into a solvable one.

Trigonometric Identities

Trigonometric identities are the backbone of simplifying complex trigonometric limit problems. These identities provide crucial formulas, allowing complex expressions to be rewritten in a more manageable form.

Consider the identity \(1 - \cos{\theta} = 2\sin^2{\left(\frac{\theta}{2}\right)}\). In the problem at hand, this identity allows us to transform \(1 - \cos{2(x-2)}\) into a form involving sin functions: \(2\sin^2{(x-2)}\). This manipulation is essential in simplifying and eventually finding the limit.

• Trigonometric identities can break down difficult trigonometric expressions, revealing simpler forms that are easier to work with.
• Employing identities correctly allows for eliminating indeterminate forms by rearranging the expression.

Mastering key trigonometric identities streamlines solving trigonometric limits significantly, transforming complex calculations into more straightforward, solvable problems.

Square Root Simplification

Simplifying expressions under a square root is an important step in many calculus problems, including those involving limits. When complicated expressions involve square roots, steps must be taken to simplify them, allowing effective evaluation.

In our example, we dealt with the square root expression \(\sqrt{2\sin^2(x-2)}\). By simplifying it to \(\sqrt{2} |\sin(x-2)|\), we expose the essential components of the equation necessary for limit evaluation.

• The simplification process often includes recognizing the properties of square roots and how they interact with squaring functions.
• Understanding these properties can aid in reducing a square root expression to a form that's compatible with trigonometric transformation, thereby removing any potential indeterminate aspect.

Mastering how to handle square root calculations within trigonometric limits prepares one for tackling more advanced calculus problems.