UPSC NDA I 2025 Mathematics Paper

Given,
The function is f(x)=x2−1 and we are tasked with finding

lim

x⟶1

(f∘f(x)).
f∘f(x)=f(f(x))
Given f(x)=x2−1, we substitute f(x) into itself:
f(f(x))=f(x2−1)
Now, apply the function f to (x2−1) :
f(x2−1)=(x2−1)2−1
This simplifies to:
f(x2−1)=x4−2x2+1−1=x4−2x2
Thus, f∘f(x)=x4−2x2.
Now, we need to find

lim

x⟶1

(x4−2x2).
Substitute x=1 into the function:
f∘f(1)=14−2(12)=1−2=−1
∴ The value of

lim

x⟶1

(f∘f(x)) is -1.