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

Question

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

Step-by-Step Solution

Answer

The limit is \(-2\).

Step 1: Calculate Trigonometric Values

To start, calculate the values of \( \sin x \) and \( \cos x \) by substituting \( x = \frac{5\pi}{4} \). At this angle, \( \sin \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) and \( \cos \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) since \( \frac{5\pi}{4} \) is in the third quadrant where both sine and cosine are negative.

Step 2: Add Trigonometric Values

Add the values obtained from Step 1: \( \sin \left(\frac{5\pi}{4}\right) + \cos \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} + -\frac{\sqrt{2}}{2} = -\sqrt{2} \).

Step 3: Evaluate the Greatest Integer Function

The greatest integer function, denoted as \([x]\), returns the largest integer less than or equal to \(x\). Here, apply this function to \(-\sqrt{2}\). Since \(-\sqrt{2} \approx -1.414 \), the greatest integer less than \(-1.414\) is \(-2\).

Step 4: State the Limit Value

Since the greatest integer less than or equal to \(-\sqrt{2}\) is \(-2\), conclude that \( \lim _{x \rightarrow \frac{5\pi}{4}} [\sin x + \cos x] = -2 \).

Key Concepts

Trigonometric Functions

When dealing with trigonometric functions like sine and cosine, it's crucial to understand how they behave based on the angle provided. The angle in this exercise, \( \frac{5\pi}{4} \), is expressed in radians, which is commonly used in calculus and trigonometry.

• Sine and Cosine Values: Both \( \sin x \) and \( \cos x \) range between -1 and 1.
• Quadrants: Angles are divided into four quadrants on the unit circle, each affecting the sign of sine and cosine.

Knowing that \( \frac{5\pi}{4} \) falls in the third quadrant is important, as both sine and cosine are negative there. This property arises because the coordinates in the third quadrant are negative for both the x and y axes, and these coordinates correspond to cosine and sine values, respectively.
In practice, a quick check with a unit circle can help determine the exact values: at \( \frac{5\pi}{4} \), \( \sin \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) and \( \cos \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \). These calculations are foundational in real-world applications like engineering and physics.

Greatest Integer Function

The greatest integer function, often denoted by \([x]\), is an essential concept for students. It is sometimes referred to as the floor function. This function rounds down a real number to the nearest integer, which is less than or equal to the number itself.

• Understanding the Concept: Think of \([x]\) as a function that 'collects' the largest piece from a fractional or decimal number without exceeding that value.
• Examples: For instance, \([3.7] = 3\) and \([-1.2] = -2\).

Applying the greatest integer function allows us to capture discrete values from continuous functions.
In this exercise, with a result of \(-\sqrt{2} \approx -1.414\), the nearest lower integer is \(-2\). This shows how we transition from a decimal result to an 'intelligible' integer, which is a crucial skill when solving real-world problems, especially in areas like computer science or economics where such transformations are often necessary.

Evaluating Limits

Evaluating limits helps bridge the understanding of how functions behave as they approach a specific point. This concept is foundational in calculus as it helps in understanding continuity, behavior of sequences, and the notion of derivatives.

• Definitions: A limit examines how a function's value approaches a particular value as its input comes near a certain point.
• Notations and Calculations: Limits can be expressed in various forms but often appear as \( \lim_{x \rightarrow c}f(x) \), indicating the x-value approaching c.

In this problem, the goal is to find the trigonometric sum as \( x \) approaches \( \frac{5\pi}{4} \).This can often require breaking down expressions into smaller parts. Here, that meant calculating \( \sin x \) and \( \cos x \) separately and adding them together before applying the greatest integer function. This set of steps helps ensure precision, and clarifying each part of the calculation can enhance comprehension.
Understanding limits involves more than calculations. It's about grasping the underlying behavior of the function's output around a certain point, which is essential in predicting and modeling numerous real-world situations.