1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Limits of an expression in x

  1. Jan 4, 2012 #1
    When sketching a graph I'm told to assume that the expression:

    f(x) =( e^x)/x

    Tends towards the infinite as x tends towards the infinite. Can someone show me how to check this?

  2. jcsd
  3. Jan 4, 2012 #2
    The short answer is that the exponential function [itex]a^x[/itex] increases faster than any power of [itex]x[/itex] ([itex]x^{\alpha}, \ \alpha \in \mathbb{R}[/itex]).

    The long answer is that you could prove that the limit [itex]\displaystyle \lim_{x\to\infty} \frac{a^x}{x^{\alpha}}[/itex] (and thus that your given limit tends towards inf) tends towards infinity for any [itex]a > 0[/itex] and [itex]\alpha \in \mathbb{R}[/itex] by expanding [itex]a^x = (1+p)^x \geq (1+p)^n[/itex], where [itex]p > 1[/itex] and [itex]n[/itex] is the integer part of [itex]x[/itex], and then doing some algebra. You should have come up with an expression which is smaller than [itex]\frac{a^x}{x^{\alpha}}[/itex] which tends to infinity, which implies the wanted result.
  4. Jan 4, 2012 #3
    Thanks, very clear.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook