Question

If \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}=2\), then \(\lim _{x \rightarrow 0}[f(x)]\) equals, (where [.] denotes the greatest integer function) (1) 2 (2) 1 (3) 0 (4) limit does not exist

Short answer

0

Step-by-step solution

- Understand the given limit

We are given that \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}=2\). This limit tells us how \(\frac{f(x)}{x^2}\) behaves as \(x\) approaches 0.

- Express the limit in terms of \(f(x)\)

From the given limit, \(\lim _{x \rightarrow 0} \frac{f(x)}{x^2} = 2\), we can express it as: \(\lim_{x \rightarrow 0} f(x) = 2x^2\).

- Evaluate the limit of \(f(x)\) as \(x\) approaches 0

Substitute \(x\) with 0 in \(2x^2\): \[ \lim_{x \rightarrow 0} f(x) = 2 \times (0^2) = 2 \times 0 = 0 \]

- Apply the Greatest Integer Function

The greatest integer function \([.]\) returns the greatest integer less than or equal to a given number. Since \(\lim_{x \rightarrow 0} f(x) = 0\), the greatest integer less than or equal to 0 is 0. Therefore, \([\lim_{x \rightarrow 0} f(x)] = 0\).

Key concepts

IIT JEE Calculus Problem

The IIT JEE examination often includes questions on calculus that require critical thinking and a good grasp of fundamental concepts. This particular problem combines the concepts of limit evaluation and the greatest integer function.
When solving such problems:

• First clearly identify the limit you need to evaluate.
• Express the given function in a simpler form to make limit evaluation straightforward.
• Understand the role of any special functions, like the greatest integer function in this case.

By following these steps, you can systematically approach and solve complex calculus problems that are commonly part of the IIT JEE examination.