Problem 69 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\lim _{x \rightarrow \pi / 3} \frac{\cos \left(x+\frac{\pi}{6}\right)}{(1-2 \cos x)^{2 / 3}}=\) (A) 1 (B) \(-1\) (C) 0 (D) None of these

Step-by-Step Solution

Answer

The limit is (D) None of these.

Step 1: Substitute the Limit

First, substitute \(x = \frac{\pi}{3}\) directly into the expression to see if the function is defined at this point. We have \(\cos\left(\frac{\pi}{3} + \frac{\pi}{6}\right)\) in the numerator and \((1 - 2\cos\left(\frac{\pi}{3}\right))^{2/3}\) in the denominator.

Step 2: Calculate Cosine Values

Calculate \(\cos\left(\frac{\pi}{3} + \frac{\pi}{6}\right) = \cos\left(\frac{\pi}{2}\right) = 0\). This leads to a numerator of zero. For the denominator, find \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) and hence \(1 - 2\cdot\frac{1}{2} = 0\). This makes the denominator also zero.

Step 3: Analyze the Indeterminate Form

Since both the numerator and the denominator become zero when \(x = \frac{\pi}{3}\), we have a \(\frac{0}{0}\) indeterminate form. Use L'Hôpital's Rule to resolve this indeterminate form.

Step 4: Apply L'Hôpital's Rule

Differentiate the numerator and the denominator with respect to \(x\). The derivative of the numerator is \(-\sin\left(x + \frac{\pi}{6}\right)\) and the derivative of the denominator, using the chain rule, is \(-\frac{2}{3} (1-2 \cos x)^{-1/3} \cdot 2 \sin x\).

Step 5: Evaluate the Limit Again

Re-evaluate the limit after differentiation using L'Hôpital's Rule: \(\lim _{x \rightarrow \pi / 3} \frac{-\sin\left(x+\frac{\pi}{6}\right)}{-\frac{4}{3}(1-2 \cos x)^{-1/3} \sin x}\)Substitute \(x = \frac{\pi}{3}\) to get \(\frac{-\sin(\frac{\pi}{2})}{-\frac{4}{3} \cdot 1 \cdot \sin(\frac{\pi}{3})}\).

Step 6: Simplify and Find the Limit

Substituting the values and simplifying gives us \(\frac{-1}{-\frac{4}{3} \cdot \frac{\sqrt{3}}{2}} = \frac{3 \sqrt{3}}{2 \cdot 4} = \frac{3 \sqrt{3}}{8}\). Thus, the limit is \(\frac{3 \sqrt{3}}{8}\).

Key Concepts

L'Hôpital's rule

L'Hôpital's Rule is an essential tool in calculus to evaluate limits involving indeterminate forms, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). When you substitute a value into a function and result in these forms, calculating the limit directly becomes impossible. Here, L'Hôpital's Rule simplifies the process.

To apply L'Hôpital's Rule, both the numerator and the denominator of your function must be differentiable near the point of interest. You take the derivative of the numerator and the derivative of the denominator separately with respect to the variable. Once you have these derivatives, you evaluate the limit again.

The rule relies on the fact that differentiation is a local linear approximation. It allows you to focus on the "rate of change" of the numerator and denominator rather than the values themselves. It is important to carefully follow through the steps of taking derivatives and re-evaluating the limit until a definite value is reached.

indeterminate forms

Indeterminate forms occur in calculus when a limit produces an ambiguous result. These forms are like mathematical puzzles that need careful solving. The most common of these forms include:

• \(\frac{0}{0}\)
• \(\frac{\infty}{\infty}\)
• \(0 \cdot \infty\)
• \(\infty - \infty\)
• \(0^0\)
• \(1^\infty\)
• \(\infty^0\)

When you encounter such cases while solving a limit problem, direct substitution doesn't work, as evidenced by your original exercise where the substitution led to \(\frac{0}{0}\). These forms indicate a lack of straightforward interpretation for a limit without further analysis.

To resolve indeterminate forms, you may use L'Hôpital's Rule or other strategies like algebraic manipulation, series expansion, or approaching the problem graphically. Understanding indeterminate forms is crucial for successfully applying calculus to seemingly unsolvable problems.

trigonometric limits

Trigonometric limits are often encountered in calculus, particularly when dealing with periodic functions such as sine and cosine. These functions have well-known specific limit properties that can simplify complex problems. When dealing with trigonometric limits, keep the following tips in mind:

• Understand key limit properties involving sine and cosine, such as \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).
• Use trigonometric identities to simplify expressions before evaluating limits.
• Take note of angles in radians, especially common angles like \(\pi/6\), \(\pi/3\), and \(\pi/2\), since their cosine and sine values are often used in exercises.

In the exercise given, calculating the trigonometric limits involved substituting \(x = \pi/3\) into cosine functions, resulting in indeterminate forms. Recognizing these patterns is essential for effectively using L'Hôpital's Rule and other methods to evaluate these types of limits efficiently.