Problem 96 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\lim _{x \rightarrow 0}\left[\frac{x^{2}}{\sin x \tan x}\right]\), where \([\cdot]\) denotes the greatest integer function, is (A) 0 (B) 1 (C) 2 (D) Does not exist

Step-by-Step Solution

Answer

The answer is (B) 1.

Step 1: Understand the Problem

We need to find the limit of the expression \( \lim_{x \to 0} \left[ \frac{x^2}{\sin x \tan x} \right] \). The square brackets \([\cdot]\) denote the greatest integer function, which means we will evaluate the limit first and then find the greatest integer less than or equal to the result.

Step 2: Apply Limit to the Expression

First, let's find \( \lim_{x \to 0} \frac{x^2}{\sin x \tan x} \). It helps to recall the standard small-angle approximations: \( \sin x \approx x \) and \( \tan x \approx x \) when \( x \to 0 \). This gives us \( \frac{x^2}{\sin x \tan x} \approx \frac{x^2}{xx} = 1 \).

Step 3: Greatest Integer Evaluation

Since the limit \( \lim_{x \to 0} \frac{x^2}{\sin x \tan x} \) evaluates to 1 as \( x \to 0 \), using the greatest integer function \([ 1 ] = 1\) because the greatest integer less than or equal to 1 is 1.

Step 4: Conclusion

Combine the steps to conclude that the expression \( \lim_{x \to 0} \left[ \frac{x^2}{\sin x \tan x} \right] = 1 \). Hence, the correct choice is (B) 1.

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted as \([ \cdot ]\), is a mathematical function that maps a real number to the largest integer less than or equal to that number. For example, \([3.9] = 3\), because 3 is the greatest integer less than 3.9. Similarly, for negative values, \([-1.2] = -2\), as -2 is the largest integer less than or equal to -1.2.

This function is also referred to as the floor function, written as \(\lfloor x \rfloor\). It is particularly useful in scenarios involving limits, such as evaluating \(\lim_{x \to 0} \left[ \frac{x^2}{\sin x \tan x} \right]\) in the given example.

• When a limit evaluates to an integer, the greatest integer function will simply yield that integer.
• If the result of a limit is not an integer, the function effectively "rounds down" to the nearest whole number.

Understanding this concept ensures precise calculations, especially in limit computations involving approximations or indeterminate forms.

Small-Angle Approximation

The small-angle approximation is a technique used in mathematics to simplify trigonometric expressions when the angle is close to zero. It is based on the premise that as the angle becomes very small, some trigonometric functions behave very predictably.

For very small angles \( x \) (measured in radians), it is generally valid to approximate:

• \(\sin x \approx x\)
• \(\tan x \approx x\)

These approximations are particularly handy when dealing with limits involving trigonometric functions. For instance, in the problem \(\lim_{x \to 0} \frac{x^2}{\sin x \tan x}\), we use the approximation to simplify the expression to \( \approx \frac{x^2}{xx} = 1 \).

This simplification is valid because as \( x \) approaches zero, the terms \(\sin x\) and \(\tan x\) closely approximate the value of \( x \) itself, making the limit more straightforward to calculate.

Indeterminate Forms

Indeterminate forms appear in calculus when attempting to evaluate the limit of a function, but the direct substitution does not lead to a clear result. Common indeterminate forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), and \(0^0\), among others.

In the problem statement provided, evaluating \(\lim_{x \to 0} \frac{x^2}{\sin x \tan x}\) could initially seem like an indeterminate form, specifically of the type \(\frac{0}{0}\), since both the numerator \(x^2\) and the denominator \(\sin x \tan x\) approach zero.

To resolve such forms, techniques like small-angle approximations, L’Hôpital’s rule, or algebraic manipulation can be used. In our case, recognizing the small-angle approximation allowed us to circumvent the indeterminate form and compute the limit as a determinate one.

• Identifying indeterminate forms is crucial for correct limit evaluations.
• Techniques to handle indeterminate forms convert them into expressions that can be straightforwardly evaluated.

Being aware of these forms and knowing how to tackle them allows for more advanced and accurate problem-solving in calculus.