Problem 1 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\lim _{x \rightarrow 0}\left(\left[\frac{100 x}{\sin x}\right]+\left[\frac{99 \sin x}{x}\right]\right)\), where \([-]\) represents greatest integer function, is (A) 199 (B) 198 (C) 0 (D) None of these

Step-by-Step Solution

Answer

The value of the limit is 198.

Step 1: Understand the limit expression

We need to evaluate the limit \( \lim_{x \rightarrow 0} \left( \left[\frac{100x}{\sin x}\right] + \left[\frac{99\sin x}{x}\right] \right) \). The brackets \([ ]\) denote the greatest integer function.

Step 2: Analyze the first term

For the term \( \frac{100x}{\sin x} \), using the small angle approximation, \( \lim_{x \to 0} \frac{x}{\sin x} = 1 \). Therefore, \( \lim_{x \to 0} \frac{100x}{\sin x} = 100 \). For small values of \( x \), \( \frac{100x}{\sin x} \) is slightly more than 100, so \( \left[ \frac{100x}{\sin x} \right] = 100 \).

Step 3: Analyze the second term

For the term \( \frac{99\sin x}{x} \), again using the approximation, \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). Therefore, \( \lim_{x \to 0} \frac{99\sin x}{x} = 99 \). For small values of \( x \), \( \frac{99\sin x}{x} \) is slightly less than 99, so \( \left[ \frac{99\sin x}{x} \right] = 98 \).

Step 4: Combine the results

Now add the results from Step 2 and Step 3: \( 100 + 98 = 198 \).

Key Concepts

Limits

The concept of limits is fundamental in calculus. A limit describes the value that a function approaches as the input approaches some value. For example, the expression \( \lim_{x \to 0} \frac{x}{\sin x} = 1 \) is saying that as \( x \) gets closer and closer to 0, the value of \( \frac{x}{\sin x} \) gets ever closer to 1. Limits help us understand behavior at points that are not clearly defined by simple evaluation.

Key steps when working with limits include:

• Substitute the approaching value into the function.
• Observe the behavior when substitution results in an indeterminate form (like \( \frac{0}{0} \)).
• Employ techniques such as small angle approximations in trigonometry to further analyze the expression.

Understanding limits allows us to predict and calculate behavior of functions, such as finding specific values a function approaches, even when direct calculation is impossible or undefined.

Greatest Integer Function

The greatest integer function is an essential mathematical tool often appearing in limit problems. Denoted by brackets \([ \cdot ]\), it takes a real number and returns the largest integer less than or equal to that number. For instance, \([2.9] = 2\) and \([-1.2] = -2\). This function introduces a critical step when working with limits as it may change the outcome when values slightly deviate from integers.

Some things to remember about the greatest integer function:

• The function is denoted by the floor brackets \([x]\).
• You are oftentimes dealing with expressions right above or below an integer, which creates a unique challenge.
• It is especially important with limits as the precise limit value determines which integer is relevant.

In limit problems, this function demands care and understanding, as small changes around points of interest can result in different integer outcomes, as seen in the solution where small deviations were critical.

Trigonometric Limits

Trigonometric limits, particularly the well-known identity \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), are vital in calculus, especially when examining behavior as things approach zero. These identities allow mathematicians to simplify expressions and understand behavior at core values, making them indispensable when solving problems involving angles or periodic functions.

Keep in mind:

• The identity \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) helps when dealing with very small angles.
• The small angle approximation \( \sin x \approx x \) is often used to ease calculations.
• Combining these identities with limits allows simplification of complex expressions.

In complex expressions, recognizing when and how to apply trigonometric limits grants deeper insights and solutions, as demonstrated in the original exercise where simplification around \(x \to 0\) was key.