Problem 87 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\lim _{x \rightarrow 5 \pi / 4}[\sin x+\cos x]\), where \([\cdot]\) denotes the greatest integer function, is (A) 2 (B) \(-2\) (C) 1 (D) \(-1\)

Step-by-Step Solution

Answer

The value is (B) -2.

Step 1: Understand the Problem

We need to find the limit of the function \( \sin x + \cos x \) as \( x \to \frac{5\pi}{4} \) and then apply the greatest integer function to the result.

Step 2: Evaluate Trigonometric Functions

Calculate \( \sin x \) and \( \cos x \) at \( x = \frac{5\pi}{4} \). At this angle, \( \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) and \( \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).

Step 3: Add the Trigonometric Values

Sum the values \( \sin\left(\frac{5\pi}{4}\right) \) and \( \cos\left(\frac{5\pi}{4}\right) \): \[\sin\left(\frac{5\pi}{4}\right) + \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} + -\frac{\sqrt{2}}{2} = -\sqrt{2}\].

Step 4: Apply the Greatest Integer Function

Find the greatest integer less than or equal to \(-\sqrt{2}\). Since \(-\sqrt{2} \approx -1.41\), the greatest integer less than or equal to this value is \(-2\).

Step 5: Conclusion and Answer Selection

The value of the limit is \(-2\), which corresponds to option (B).

Key Concepts

Trigonometric Functions

Trigonometric functions play a crucial role in calculus, especially when dealing with limits and periodic functions. The basic trigonometric functions, sine (\( \sin \)) and cosine (\( \cos \)), are defined on the unit circle. Their output values vary from -1 to 1 as they describe the coordinates of points on the circle.

• At specific angles, such as \( \frac{\pi}{4} \), these functions take known values. For instance, \( \sin \left( \frac{5\pi}{4} \right) \) and \( \cos\left(\frac{5\pi}{4}\right)\) both equal \(-\frac{\sqrt{2}}{2}\).
• This is due to the symmetry of the unit circle; the angle \(\frac{5\pi}{4}\) situates in the third quadrant where both sine and cosine are negative.

Understanding these trigonometric values and their sums, such as in \( \sin x + \cos x \), is essential when applying functions like the greatest integer function.

Greatest Integer Function

The greatest integer function, also known as the floor function, returns the largest integer less than or equal to a given number. Mathematically, it is represented by \([x]\), meaning it takes a real number \(x\) and "rounds down" to the nearest integer.

• For example, \([3.7] = 3\) and \([-1.41] = -2\).
• In our exercise, when dealing with \(-\sqrt{2} \approx -1.41\), applying this function yields \([-\sqrt{2}] = -2\).

This rounding down process is useful in problems where discrete values are needed, such as calculating precise limits in calculus.

Calculus Problem Solving

In calculus, problem solving often involves understanding how different mathematical concepts come together. Let's break it down with our example involving the limit and greatest integer function.

• The first step is to evaluate or approximate function values, such as finding \( \sin x \) and \( \cos x \) at specific points.
• Next, sum these evaluated trigonometric expressions if required by the problem statement, as we did with \( \sin\left(\frac{5\pi}{4}\right) + \cos\left(\frac{5\pi}{4}\right) = -\sqrt{2}\).
• Finally, apply additional functions or operations, like the greatest integer function, to derive your final answer. This function demanded us to determine \([-\sqrt{2}] = -2\), the concluding step in the evaluation.

Calculus problem solving is a journey that requires careful evaluation, thoughtful application of mathematical operations, and often a bit of intuition to interpret results correctly. The process is essential for finding rates of change, integrals, limits, and other core calculus operations.