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Proofs on Limit and Derivatives

  1. May 8, 2007 #1
    1) Prove that f defined by
    f(x)= e^(-1/|x|), x=/=0,
    f(x)= 0, x=0
    is differentiable at 0.

    [I used the definition of derivative
    f'(0)=lim [f(0+h)-f(0)] / h = lim [e^(-1/|h|) / h]
    h->0 h->0
    and I am stuck here and unable to proceed...]

    2) Suppose lim (x->a) f(x) = L exists and f(x)>0 for all x not =a. Use the definition of limit to prove that L>0.

    [when I draw a picture, I can see that this is definitely true, but how can I go about proving it?]

    Thanks for your help!:smile:
    Last edited: May 8, 2007
  2. jcsd
  3. May 8, 2007 #2


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    For the second one, start with the definition of limit. Suppose L is less than zero, and show there must be some f(x) for x near a such that f(x) is quite close to L, and hence negative (in more formal terms of course)
  4. May 8, 2007 #3


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    Let k= 1/h so that you have
    [tex]\lim_{k\rightarrow \infty} ke^-k[/itex]
    and use L'Hopital's rule.

    You mean "limit as x goes to a of f(x)", not " limit of (x-a)f(x)" surely, since in the second case this is not true. Use 'indirect proof'. Suppose L< 0 and take [itex]\epsilon[/itex]= L/2 in the definiton of limit.
  5. May 8, 2007 #4
    1) But there is an absolute value |h|, letting k=1/h won't get rid of the absolute value, right?

    2) Sorry, that's a typo...I have corrected it...
    How do you know how epsilon you are going to pick? I am personally having terrible trouble knowing what epsilon to pick to do this kind of proofs...
  6. May 8, 2007 #5
    Can someone please help me? I will be writing my finals tomorrow...
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