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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.
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