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

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

(A) \(\cos^4 \theta \ln \cos \theta - \sin^4 \theta \ln \sin \theta\)

Step 1: Identify the form of the limit

The given limit is of the form \( \frac{f(x) - f(4)}{x - 4} \), which resembles the definition of the derivative of \( f(x) \) at \( x = 4 \).

Step 2: Define the function

Let \( f(x) = \cos^x \theta - \sin^x \theta - \cos 2\theta \). Then the expression becomes \( \lim_{x \to 4} \frac{f(x) - f(4)}{x - 4} \). We need to find \( f'(4) \).

Step 3: Differentiate \( f(x) \) with respect to \( x \)

The derivative \( f'(x) \) of \( f(x) = \cos^x \theta - \sin^x \theta - \cos 2\theta \) is obtained using the property that \( \frac{d}{dx} a^x = a^x \ln a \). Thus, \( f'(x) = \cos^x \theta \ln \cos \theta - \sin^x \theta \ln \sin \theta \).

Step 4: Evaluate the derivative at \( x = 4 \)

Substituting \( x = 4 \) in the derivative, we have \( f'(4) = \cos^4 \theta \ln \cos \theta - \sin^4 \theta \ln \sin \theta \).

Step 5: Compare with the given options

The expression \( \cos^4 \theta \ln \cos \theta - \sin^4 \theta \ln \sin \theta \) matches option (A).

Key Concepts

Derivative

The concept of a derivative is central to calculus and provides a way to measure how a function changes as its input changes. You can picture the derivative as the "slope" of the function at any given point. If you've ever ridden a bike up a hill, you've experienced the concept of a slope firsthand. In calculus, we use derivatives to describe how steep the hill is at every point along the way.

When we compute the derivative of a function, we typically start with the limit definition: the derivative of a function \( f(x) \) at a point \( x = a \) is given by:\[ f'(a) = \lim_{x\to a} \frac{f(x) - f(a)}{x - a}\]This expression shows how we analyze the change in the function value relative to the change in \( x \). This is exactly what we were asked to explore in the exercise by finding the derivative at \( x = 4 \).

The result shows how different rates of change are for each part of the function, combining the trig functions and the subtraction of constants. Utilizing the power of derivatives allows us to understand the nuances of functions more deeply, turning abstract concepts into tangible, understandable slopes.

Trigonometric Functions

Trigonometric functions, such as cosine and sine, are fundamental in both geometry and calculus, connecting angles and side lengths in right-angled triangles to circular motion. These functions are periodic, which means they repeat values in regular intervals, embodying waves' behavior in physics, sound, and light.

In the context of calculus, trigonometric functions are interesting because their derivatives also produce trigonometric or constant functions. For instance:

• Derivative of \( \cos(x) \) is \( -\sin(x) \).
• Derivative of \( \sin(x) \) is \( \cos(x) \).

These properties make them useful in a wide range of applications, from solving differential equations to modeling periodic behavior.

In our exercise, \( \cos \theta \) and \( \sin \theta \) are raised to the power of \( x \), coupling their periodic nature with the power rule of derivatives, demonstrating the versatility and interconnected nature of mathematics. This seamless blending shows the elegance of trig functions integrating into the broader mathematical tapestry.

Continuous Functions

Continuous functions are a basic yet profound concept in calculus. A function is continuous if you can draw its graph without lifting your pencil from the paper; at every point on the graph, there are no breaks, jumps, or holes. This idea is critical because many calculus techniques assume continuity. For instance, both derivatives and integrals typically require continuous functions.

For a function \( f(x) \) to be continuous at a point \( x = c \), it must satisfy:

• \( f(c) \) is defined.
• \( \lim_{x\to c} f(x) = f(c) \).
• The limits from the left and right match: \( \lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) \).

In the exercise provided, ensuring the continuity of parts like \( \cos^x \theta - \sin^x \theta \) across all values \( x \) is necessary for applying derivative rules correctly. Understanding continuous functions allows mathematics to be applied smoothly across real-world problems and abstract thinking, offering a reliable bridge between numerical concepts and practical applications.