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

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SUMMARY

The limit statement lim{x-> infinity} f(x)/x = infinity directly implies that lim{x-> infinity} f(x) = infinity due to the growth rate of f(x) relative to x. Specifically, if f(x) grows faster than x, then as x approaches infinity, f(x) must also approach infinity. Any assumption that lim f(x) is finite or does not exist leads to a contradiction, reinforcing the necessity of f(x) tending to infinity.

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Homework Statement



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

What is evidence?

Thank you very much.
 
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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.
 

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