Problem 75 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\lim _{x \rightarrow 4} \frac{(\cos \theta)^{x}-(\sin \theta)^{x}-\cos 2 \theta}{x-4}=\) (A) \(\cos ^{4} \theta \ln \cos \theta-\sin ^{4} \theta \ln \sin \theta\) (B) \(\cos ^{4} \theta \ln \cos \theta+\sin ^{4} \theta \ln \sin \theta\) (C) \(\cos ^{4} \theta \ln \sin \theta-\sin ^{4} \theta \ln \cos \theta\) (D) None of these

Step-by-Step Solution

Answer

The answer is (A): \(\cos ^{4} \theta \ln \cos \theta - \sin ^{4} \theta \ln \sin \theta\).

Step 1: Recognize Undefined Form

Evaluate the given limit: \( \lim _{x \rightarrow 4} \frac{(\cos \theta)^{x}-(\sin \theta)^{x}-\cos 2\theta}{x-4} \). Since direct substitution results in a \(\frac{0}{0}\) form, we need to apply L'Hôpital's Rule or another method.

Step 2: Apply L'Hôpital's Rule

When a limit is in the \(\frac{0}{0}\) form, L'Hôpital's Rule can be applied, which requires differentiating the numerator and the denominator. Differentiate the numerator, \((\cos \theta)^{x}-(\sin \theta)^{x}-\cos 2\theta\), with respect to \(x\), treating \(\theta\) as a constant.

Step 3: Differentiate the Expression

The derivatives are as follows: \(\frac{d}{dx}\left((\cos \theta)^{x}\right) = (\cos \theta)^{x} \ln \cos \theta\) and \(\frac{d}{dx}\left((\sin \theta)^{x}\right) = (\sin \theta)^{x} \ln \sin \theta\). The derivative of the constant \(-\cos 2\theta\) is zero. Thus, the numerator becomes \((\cos \theta)^{x} \ln \cos \theta - (\sin \theta)^{x} \ln \sin \theta\).

Step 4: Apply the Derivatives

Next, apply these derivatives in L'Hôpital's Rule: \[ \lim _{x \rightarrow 4} \frac{(\cos \theta)^{x} \ln \cos \theta - (\sin \theta)^{x} \ln \sin \theta}{1} \].

Step 5: Substitute the Limit

Since this is no longer an indeterminate form, replace \(x\) with 4: \((\cos \theta)^{4} \ln \cos \theta - (\sin \theta)^{4} \ln \sin \theta\).

Step 6: Identify the Answer

The expression obtained after substitution matches option (A): \(\cos ^{4} \theta \ln \cos \theta - \sin ^{4} \theta \ln \sin \theta\).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a valuable tool in calculus, particularly when you're dealing with limits that result in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When you come across these forms, direct substitution won't give a meaningful answer. That's where L'Hôpital's Rule comes in handy.
By differentiating the numerator and the denominator separately, you convert the original problem into a simpler one. If the new limit is still indeterminate, you can apply L'Hôpital's Rule again. It's important to note that L'Hôpital's Rule is only applicable under certain conditions.

• The original limit must be in an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• The functions in the numerator and denominator need to be differentiable.
• The limit of the derivatives must exist or also lead to an indeterminate form.

In the given exercise, L'Hôpital's Rule helps us find the limit of the function involving trigonometric powers and simplifies the complex nature of the expression.

Derivatives

Derivatives are a fundamental aspect of calculus. They represent the rate of change of a function concerning one of its variables. In simpler terms, a derivative shows how a function's output changes as you change its input.
There are different rules and techniques to find derivatives, such as the power rule, product rule, quotient rule, and chain rule. In this particular exercise, derivatives of exponential functions with respect to a variable are important. The function derivatives are \( \frac{d}{dx}\left((\cos \theta)^{x}\right) = (\cos \theta)^{x} \ln \cos \theta \) and \( \frac{d}{dx}\left((\sin \theta)^{x}\right) = (\sin \theta)^{x} \ln \sin \theta \).
Here, the derivatives involve the natural logarithm because of the variable exponent. Understanding how to differentiate such functions is crucial when applying L'Hôpital's Rule. This knowledge helps simplify expressions, making limits more approachable.

Indeterminate Forms

Indeterminate forms arise when direct substitution in a limit equation gives results like \( \frac{0}{0} \), \( \infty - \infty \), or \( \frac{\infty}{\infty} \). These forms do not provide a direct answer and often require more advanced techniques like L'Hôpital's Rule to resolve.
Understanding why a form is indeterminate is essential. It tells you that the usual process of substitution won't work, and you'll need to find a way to transform the expression. Recognizing these forms allows you to choose the right method to solve the limit problem.

• The most common indeterminate forms that require techniques like L'Hôpital's Rule to solve include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), and multiple others like \( 0 \cdot \infty \) or \( \infty^0 \).

In the given exercise, realizing that the expression \((\cos \theta)^{x}-(\sin \theta)^{x}-\cos 2\theta\) over \(x-4\) gives a \(\frac{0}{0}\) form is pivotal. It indicates the need for differentiating to make the evaluation possible.