UPSC NDA I 2025 Mathematics Paper

The function is defined as:
f(x)={

x3,‌	‌ if ‌|x|<1
x2,‌	‌ if ‌|x|≥1

We are tasked with finding:

lim

x⟶0

f′(x)
For |x|<1, the function is f(x)=x3, so the derivative is:
f′(x)=3x2
Now, compute the limit of the derivative as x to 0 :

lim

x⟶0

f′(x)=

lim

x⟶0

3x2=0
∴ The value of

lim

x⟶0

f′(x) is 0 .