What is \(\lim_{x \to 0} f'(x)\) equal to?

Question

Question

Consider the following for the two (02) items that follow:

\(\text{Let } f(x)= \begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \ge 1 \end{cases} \\\)

This question was previously asked in

NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)

Answer 1

Calculation:

Given,

The function is defined as:

\( f(x) = \begin{cases} x^3, & \text{if} \, |x| < 1 \\ x^2, & \text{if} \, |x| \geq 1 \end{cases} \)

We are tasked with finding:

\( \lim_{x \to 0} f'(x) \)

For |x| < 1 , the function is f(x) = x³, so the derivative is:

\( f'(x) = 3x^2 \)

Now, compute the limit of the derivative as x to 0:

\( \lim_{x \to 0} f'(x) = \lim_{x \to 0} 3x^2 = 0 \)

∴ The value of \(\lim_{x \to 0} f'(x) \)is 0.

The correct answer is Option (c)

Download Solution PDF