Re: Compute the limit of x/exp(x) when x approaches positive infinite?
From: Julian V. Noble (jvn_at_virginia.edu)
Date: 08/10/04
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Date: Tue, 10 Aug 2004 16:20:11 -0400
Daniel Mark wrote:
>
> Hello all:
>
> I want to compute the limit of formula x/exp(x) when x approaches positive
> infinite.
>
> I use the following procedure to prove that it is 0.
> Is this method correct?
>
> 1> 0 <= x/exp(x) <= x/x^2 = 1/x when x approaches positive infinite.
>
> 2> 0 <= x/exp(x) <= 1/x = 0 when x approaches positive infinite.
>
> 3> therefore, x/exp(x) ==> 0
>
> thank you
> -Daniel
You could use L'Hopital's Rule, which is probably what your instructor
intended. See Grossman, Calculus (3rd ed, ch. 12) or search Google online.
For example,
http://www.ltcconline.net/greenl/courses/106/approxother/LHOP.HTM
has a reasonably clear explanation based on the Mean Value Theorem. Keep
in mind that you can only use L'Hopital's Rule to find the limit of f(x)/g(x)
when both functions go to 0 or infinity. If one remains finite [or bounded,
as in the case cos(x)/ln(x) ] you don't get the right limit.
--
Julian V. Noble
Professor Emeritus of Physics
jvn@lessspamformother.virginia.edu
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
"For there was never yet philosopher that could endure the toothache
patiently." -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.
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