Limit IV

Rushil

If $f(x)$ be a differentaible function such that $\lim_{x \longrightarrow \infty} f(x) = a \in \mathbb{R}$ . Suppose that $\lim_{x \longrightarrow \infty} x f'(x)$ exists. Find this limit.

**Posted:** Thu Sep 22, 2005 11:08 pm

liyi

The limit $\lim_{x\to\infty} xf'(x)$ must be zero. If not, WLOG, suppose it is a positive real. Then there exists $X>1$ such that $xf'(x)>a>0$ for $x>X$ . $f'(x)>a/x$ for $X>0$ . Choose $b>X$ .

Let us prove that $f(x)>f(b)+a\ln x-a\ln b$ for $x>b$ .
To see this, let $g(x) = f(x)-f(b)-a\ln x+a\ln b$ . In fact, $g'(x) = f'(x) - \frac{a}{x} > 0$ for $x>b$ and $g(b)=0$ . Thus $g(x)>0$ for $x>b$ .

$f(x)>f(b)+a\ln x-a\ln b$ implies that $f$ is unbounded. Contradiction.

**Posted:** Fri Sep 23, 2005 12:27 am

Rushil

thanks!!! Good proof!

**Posted:** Fri Sep 23, 2005 1:55 am

liyi

A small comment: A more natural thought is use $\int_b^x f'(x)dx > \int_b^x \frac{a}{x}dx$ to get the contradiction. But is $f'(x)$ integrable? We need some words to argue. So I use derivatives instead of integration in the proof.

**Posted:** Fri Sep 23, 2005 7:18 am