WHY lim{x-> infinity} f(x)/x = infinity => lim{x-> infinity} f(x) = infinity ?

1. Nov 13, 2009

gymko

1. The problem statement, all variables and given/known data

Why is it?
lim{x-> infinity} f(x)/x = infinity => lim{x-> infinity} f(x) = infinity

What is evidence?

Thank you very much.

2. Nov 13, 2009

Staff: Mentor

The reason this is so is that the first limit being infinity implies that f(x) is growing at a faster rate than x is.

As for evidence, if you assume that lim f(x) is finite or doesn't exist, then lim [f(x)/x] is 0 or doesn't exist as x gets large, which is a contradiction. If you were going to prove this, you would have to be a little more careful, but maybe you get the idea I'm trying to convey.