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

  1. Nov 13, 2009 #1
    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. jcsd
  3. Nov 13, 2009 #2

    Mark44

    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.
     
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