- Here's the question :
Let $f : R\to R$ be a real function. The function $f$ is derivable and there exists $n \in N$ and $p \in R$ such that $\lim_{x \to \infty} x^nf(x) = p$, and there exists $lim_{x \to \infty} (x^{n+1}.f(x))$ , then evaluate $\lim_{x \to \infty} (x^{n+1}f'(x))$
- What I did :
The limit which equals $p$ can be solved using L' Hopital's Rule (once or many time depending on $f(x)$ and $p$)
So applying L' Hopital's Rule once to the term in the limit will yield the same value of the limit.
$$\lim_{x \to \infty} \frac{f(x)}{x^{-n}} = p$$
Apply L-H Rule here
$$\lim_{x \to \infty} \frac{f'(x)}{-nx^{-(n+1)}} = p$$
$$\lim_{x \to \infty } f'(x)x^{n+1} = -np$$
Now I am looking for alternate ways to do this question
So please mention how would you do this problem; Thanks...