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!

Limit as x-> inf

Tags:
  1. Feb 16, 2017 #1
    1. The problem statement, all variables and given/known data
    Find ##\lim_{x\to\infty} x(e^{1/x}-1)##

    2. Relevant equations
    ##\lim_{x\to\infty} \frac{f(x)}{g(x)} = \lim_{x\to\infty} \frac{f'(x)}{g'(x)}##

    3. The attempt at a solution
    I attempted to rewrite the function in terms of a ratio and then use L'Hopital's rule:

    ##\lim_{x\to\infty} \frac{x}{(e^{1/x}-1)^{-1}} = \lim_{x\to\infty} \frac{1}{-(e^{1/x}-1)^{-2}(\frac{1}{x}e^{1/x})}##

    The problem is that the exponential terms never go away. The bigger problem is I believe L'Hopital's Rule is probably unnecessary and I'm missing something more basic here.
     
  2. jcsd
  3. Feb 16, 2017 #2

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Would it be easier for you to investigate :smile: $$\lim_{\varepsilon\downarrow 0} {e^\varepsilon -1 \over \varepsilon } \ \ \rm ? $$
     
  4. Feb 16, 2017 #3

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Another idea is to use power series.
     
  5. Feb 17, 2017 #4
    I am able to see what you did today :oldsmile:. ##\epsilon = 1/x , x = 1/ \epsilon## Now I can infer that as x goes to infinity, epsilon goes to zero, so by making this replacement, we also replace the limit from infinity to zero. Then once we get into this nice form, L'Hopital's Rule and we're good. I'm not sure if I've ever seen this technique before, at least I don't remember it.
     
  6. Feb 17, 2017 #5

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You could write the ratio differently.
    $$\lim_{x\to\infty} \frac{e^{1/x}-1}{1/x}$$
    That form succumbs straightforwardly to the hospital rule.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Limit as x-> inf
  1. A limit as x->inf (Replies: 11)

Loading...