Problem 137 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(\lim _{x \rightarrow 0} \frac{(1-\cos 2 x)(3+\cos x)}{x \tan 4 x}\) is equal to (A) \(\frac{1}{2}\) (B) 1 (C) 2 (D) \(-\frac{1}{4}\)

Step-by-Step Solution

Answer

The limit is 2, so the answer is (C) 2.

Step 1: Apply Trigonometric Identities

First, apply trigonometric identities to simplify the expression. For the term \(1 - \cos 2x\), use the identity \(1 - \cos 2x = 2\sin^2 x\). For \(\tan 4x\), use \(\tan 4x \approx 4x\) when \(x \to 0\).

Step 2: Substitute Identities

Substitute the trigonometric identities into the expression: \[\lim _{x \rightarrow 0} \frac{(2\sin^2 x)(3+\cos x)}{x \cdot 4x}\] Simplifying this gives:\[\lim _{x \rightarrow 0} \frac{2\sin^2 x (3+\cos x)}{4x^2}\]

Step 3: Apply Limits

Divide throughout by \(x^2\) and apply limits:\[\lim _{x \rightarrow 0} \frac{2\sin^2 x}{x^2} \cdot \lim_{x \rightarrow 0} (3 + \cos x) / 4\]Using the limit \(\lim_{x \rightarrow 0} \frac{\sin^2 x}{x^2} = 1\), this becomes:\[\frac{2}{4} \cdot (3 + 1)\]

Step 4: Calculate the Final Expression

Now calculate the result:\[\frac{2}{4} \cdot 4 = \frac{8}{4} = 2\]Thus, the limit is \(2\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential in simplifying expressions that involve trigonometric functions. They are particularly useful in calculus for solving limit problems. In this exercise, knowing the identity \(1 - \cos 2x = 2\sin^2 x\) allows us to transform the expression into a more workable form.

• \(\sin^2 x\) is a standard representation of the square of the sine function, commonly occurring in many limit problems.
• The approximation \(\tan 4x \approx 4x\) for small values of \(x\) is another handy identity that aids in simplifying the denominator.

By substituting these identities, we transform complicating functions into simpler algebraic forms, making the evaluation of limits more straightforward.

Calculus Problem Solving

In calculus, problem solving is not just about getting the answer, but understanding how to navigate from complex expressions to simpler forms. The expression \((1-\cos 2x)(3+\cos x)/(x \tan 4x)\) initially seems intricate. The key is in the strategy:

• First, identify potential identities or approximations that can transform terms within the limit expression.
• Simplifying these terms using known limits can make complex calculations possible.

With these steps, we slice through initial complexity to find straightforward numerical values as we have seen in this problem. This approach highlights the interplay between trigonometry and limits in calculus.

Limit Evaluation Techniques

Evaluating limits often involves breaking down expressions by applying well-known limits and identities. The hallmark here is the understanding of common limits. Let's review:

• The limit \(\lim_{x \to 0} \frac{\sin^2 x}{x^2} = 1\) is pivotal in simplifying expressions where \(x\) approaches zero.
• Breaking the original expression into separate limit expressions allows us to individually compute each component, making the full expression easier to evaluate.

Ultimately, using these techniques, we rewrite the problem into easier forms: \( \frac{2}{4} \cdot (3 + 1) \), and compute them individually. This division into stages demonstrates the powerful utility of limit evaluation techniques, leading from complex fractional expressions to simple arithmetic solutions.