Definite Integration
Iterated Function and Limit of Integral
nta_pyq_2023_apr
Grade 12
Question:
Let $f(x)=\dfrac{x}{(1+x^n)^{1/n}}$, $x\in\mathbb{R}\setminus\{-1\}$, $n\in\mathbb{N}$, $n>2$. If $f_n(x)=(f\circ f\circ\cdots$ up to $n$ times$)(x)$, then $\displaystyle\lim_{n\to\infty}\int_0^1 x^{n-2}(f_n(x))\,dx$ is equal to
Step-by-Step Solution
Key Concept: Show $f_n(x)=\dfrac{x}{(1+nx^n)^{1/n}}$ by induction. Then compute $\int_0^1 x^{n-1}(1+nx^n)^{-1/n}dx$ and take the limit.
$\lim_{n\to\infty}I_n=0$.
Correct Answer: 0