NDA 1 2025 Maths, question 79

Topic: Differential calculus

Passage · Q79 to Q80

Let $f(x) = \left\{ \begin{matrix} x^{3}, & x^{2} < 1, \\ x^{2}, & x^{2} \geq 1. \end{matrix} \right.$

Q79

What is $\lim_{x \rightarrow 0}{f\prime(x)}$ equal to?

(a) $2$
(b) $1$
(c) $0$ Correct answer
(d) Limit does not exist

Solution.

Answer: (c) $0$

In a neighbourhood of zero, $x^{2} < 1$ , so $f(x) = x^{3}$ and $f\prime(x) = 3x^{2}$ .
Therefore $\lim_{x \rightarrow 0}{f\prime(x)} = 0$ .
The behaviour at $x = \pm 1$ does not affect this limit.