Let f(x) = x1+x^n, x R - -1, n N, n 2. If f^n(x) = n(f(f( f(x))))(x), then n 0^1 x^n-2 (f^n(x)) dx is equal to:

Let f(x) = x1+x^n, x R - -1, n N, n 2. If f^n(x) = n(f(f( f(x))))… — Solution | JEE Mathematics

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Let $f(x) = \frac{x}{1+x^n}$ , $x \in \mathbb{R} - \{ -1 \}$ , $n \in \mathbb{N}$ , $n > 2$ . If $f^n(x) = n(f(f(\cdots f(x))))(x)$ , then $\lim_{n \to \infty} \int_0^1 x^{n-2} (f^n(x)) \, dx$ is equal to:

Answer

Correct answer:0

Step-by-step solution

Standard Method

Given: $f(x) = \frac{x}{1+x^n}$ and we need to evaluate $\lim_{n \to \infty} \int_0^1 x^{n-2}(f^n(x)) \, dx$ .

Find: The limiting value of the integral.

Using limit and series expansion:

\lim_{n \to \infty} \int_0^1 x^{n-2} (f^n(x)) \, dx = 0

Therefore, the value of the given limit is $0$ .

Common mistakes

Interpreting $f^n(x)$ carelessly can lead to error. One may confuse iteration with ordinary power. Follow the notation exactly as defined in the question before evaluating the integral.
Assuming the integral has a nonzero contribution from the full interval $[0,1]$ is incorrect for large $n$ . The factor $x^{n-2}$ suppresses most contributions, so the limiting behavior must be handled carefully.

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Answer

Step-by-step solution

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