UPSC NDA I 2025 Mathematics Paper

Concept:The limit of a composite function is found by first simplifying the composition and then substituting the limit point directly, as the resulting expression is a polynomial.Explanation:Given $f(x)=x^{2}-1$. We need $\underset{x\to1}{\lim} f\circ f(x) = \underset{x\to1}{\lim} f(f(x))$.First compute $f\circ f(x)$: $f(f(x)) = f(x^{2}-1) = (x^{2}-1)^{2}-1$.Simplify: $(x^{2}-1)^{2} = x^{4}-2x^{2}+1$, so $(x^{2}-1)^{2}-1 = x^{4}-2x^{2}+1-1 = x^{4}-2x^{2}$.Thus $f\circ f(x)=x^{4}-2x^{2}$.Now find the limit as $x\to1$: substitute $x=1$ into the polynomial.$f\circ f(1)=1^{4}-2(1^{2})=1-2=-1$.Hence the limit is $-1$.Answer:$-1$, which corresponds to option A.