Problem 136 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Limit of \(\left(\frac{\sqrt{1-\cos \\{2(x-2)\\}}}{x-2}\right)\) as \(x\) tends to 2 (A) equals \(\sqrt{2}\) (B) equals \(-\sqrt{2}\) (C) equals \(\frac{1}{\sqrt{2}}\) (D) does not exist

Step-by-Step Solution

Answer

(A) equals \(\sqrt{2}\).

Step 1: Identify the Limit Expression

We are given the expression \( \lim_{x \to 2} \frac{\sqrt{1-\cos \{2(x-2)\}}}{x-2} \). Our task is to find its limit as \( x \to 2 \).

Step 2: Substitute the Variable

Set \( y = x - 2 \), which implies \( x = y + 2 \). Now, as \( x \to 2 \), \( y \to 0 \). Transform the expression to \( \lim_{y \to 0} \frac{\sqrt{1-\cos(2y)}}{y} \).

Step 3: Apply Trigonometric Identity

Recall the trigonometric identity \( 1 - \cos(x) = 2\sin^2\left(\frac{x}{2}\right) \). Apply it: \[ 1 - \cos(2y) = 2\sin^2(y) \]. This turns the expression into \( \lim_{y \to 0} \frac{\sqrt{2\sin^2(y)}}{y} = \lim_{y \to 0} \frac{\sqrt{2}|\sin(y)|}{y} \).

Step 4: Simplify Using Sine Limit

Since \( \sin(y) \approx y \) when \( y \to 0 \), replace \( |\sin(y)| \) with \( y \): \[ \lim_{y \to 0} \frac{\sqrt{2}|\sin(y)|}{y} = \lim_{y \to 0} \frac{\sqrt{2}y}{y} \].

Step 5: Evaluate the Simplified Limit

Cancel out \( y \) from the numerator and denominator: \( \lim_{y \to 0} \sqrt{2} = \sqrt{2} \).

Step 6: Conclusion

The limit is \( \sqrt{2} \), as the function simplifies to \( \sqrt{2} \) for \( y \to 0 \). Thus, the answer is (A) \( \sqrt{2} \).

Key Concepts

Trigonometric Identities

In calculus, trigonometric identities are crucial for simplifying complex expressions, especially involving limits. An identity often used is the double-angle formula for cosine:

• \( 1 - \cos(x) = 2\sin^2\left(\frac{x}{2}\right) \).

This identity allows us to transform expressions in order to simplify calculations, such as evaluating the limits of trigonometric functions.
By factoring trigonometric terms into more manipulable algebraic terms, calculations become more straightforward. In this exercise, the identity helped change the original limit expression to a simpler form, crucial for finding the limit as \( x \to 2 \). By converting \( 1-\cos(2y) \) into \( 2\sin^2(y) \), it became possible to solve the limit problem without encountering indeterminate forms.

Limits of Trigonometric Functions

Understanding how the limits of trigonometric functions work is vital in calculus. Trigonometric limits might sometimes appear complicated due to oscillatory behavior of functions like sine and cosine. However, these functions often have well-known behaviors near zero. For instance:

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

In this problem, knowing the behavior of \( \sin(y) \) near zero was essential to simplify the given expression to \(\lim_{y \to 0} \frac{\sqrt{2}|\sin(y)|}{y}\).
Since \(\sin(y)\) approximates to \(y\) as \(y\) approaches zero, it allows the transformation of complex expressions into solvable forms. Recognizing and applying these trigonometric limits can significantly simplify evaluating limits involving trigonometric expressions.

Indeterminate Forms

Indeterminate forms commonly arise in calculus when evaluating limits. One such form, \( \frac{0}{0} \), indicates that straightforward substitution does not lead to an answer.
When you see such a form, consider algebraic manipulation or applying l’Hôpital’s Rule, if applicable, which involves differentiation. However, in trigonometry, indeterminate forms often resolve themselves through identities or approximate values, as seen in this exercise.
The initial form, \( \frac{\sqrt{1-\cos(2y)}}{y} \), seemed indeterminate. By transforming \( 1-\cos(2y) \) using a trigonometric identity, it was possible to reframe the expression, leading to cancellations that resolved the indeterminacy. Always look for algebraic methods or identities, like trigonometric ones, for handling indeterminate forms before resorting to other techniques.

Function Transformation

Function transformation is a key tool in calculus for evaluating limits, simplifying functions, or changing variable scales. In this exercise, transforming the variable \(x\) by setting \(y = x - 2\) enabled the limit to be evaluated as \( y \to 0 \) instead of \( x \to 2 \).
Function transformations simplify expressions and sometimes reveal hidden symmetries or patterns that aren't easily visible initially. For example:

• Substituting \(y = x - 2\) turned the limit into a more manageable form \( \lim_{y \to 0} \), focusing on the behavior of the function at zero.

By simplifying the expression and translating the problem into a new variable, resolving problems with limits becomes more intuitive and straightforward, allowing us to focus on the behavior of simpler functions.