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!

Simple limit

  1. Apr 27, 2010 #1
    Hello, it is pretty obvious that the following limit is equal to zero:

    [tex]$Lim t \rightarrow \infty (\tmop{te}^{- t}) = 0$[/tex]

    For example, for t=100 it is [tex]100*e^{-100}[/tex]

    But how would you take this limit "rigorously"? I tried decomposing the function with a mclaurin series and [tex]te^-t[/tex] is equal to this series:

    [tex]$\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1} t^n}{(n - 1) !}$[/tex]

    How would I actually evaluate this series for t->infinity???? Or is this the wrong approach?

    Also for a finite number of terms it appears that this series diverges...
  2. jcsd
  3. Apr 27, 2010 #2
    Errrr... L'hopital's rule. Sorry should have spent a while longer thinking about it before posting.
  4. Apr 28, 2010 #3


    User Avatar
    Science Advisor

    Write it as [itex]t/e^t[/itex] and use L'Hopital's rule as Nick R suggested.
  5. May 6, 2010 #4


    User Avatar
    Science Advisor

    Nick R = TS ;)

    A more direct proof: since [itex]e^x = 1 + x + x^2+ ... [/itex], it is obvious that [itex]e^x>x[/itex] for all [tex]x\in\mathbb{R}[/tex]. In other words, [itex]\frac{e^x}{x}>1[/itex]. Hence

    [tex]\frac{e^x}{x}=\frac{1}{2}\left(\frac{e^{x/2}}{x/2}\right)e^{x/2}>\frac{1}{2}e^{x/2}\to\infty[/tex] if [itex]x\to\infty[/itex].

    It follows that [itex]xe^{-x}=\frac{x}{e^x}\to 0[/itex] if [itex]x\to\infty[/itex].
  6. May 9, 2010 #5
    Ts = op?
  7. May 9, 2010 #6


    User Avatar
    Science Advisor

    I'm sorry, with TS I meant Topic (/Thread) Starter. Is OP (original poster?) more standard?
  8. May 9, 2010 #7
    We are mathematicians. We can call it whatever we want! But it is mandatory to use at least two of these:
    1) greek letter(s)
    2) subscript
    3) AlTeRnAtInG CaPs

    I recommend that we define [tex]\tau\sigma_{1}(399107)[/tex]:= {"Nick R"}
  9. May 9, 2010 #8


    User Avatar
    Homework Helper

    Hahaha that's a good one :rofl:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook