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Proving limit equivalence statements.

  1. Oct 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Is it always true that

    lim f(x) x-->infinity = lim f(1/t)t --> 0

    lim f(x)x--> 0 = lim f(1/n)n-->infinity

    3. The attempt at a solution

    How can you begin to prove or disprove these statements if you don't know what f is doing to x. In other words, lim f(x) could not exist or it could depending on what f(x) is doing to the x's right? So from where do I start?
     
  2. jcsd
  3. Oct 20, 2009 #2

    lanedance

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    not too sure on this one, but as some ideas, consider the 2nd case
    lim x->0, f(x)

    for the limit to exist, the limit must be same for the approach to zero form both sides
    lim x->0+, f(x) = lim x->0-, f(x)

    it seems to me changing to the case, lim n-> inf, f(1/n) only really consider the 0+ approach

    consider a step function at the origin

    however that will only be a problem if the limit does not exist, if the limit does exist then i think you might be ok - however it might be some food for though for the first case what if t approaches 0-, then would you require the negative infinte limit to be the same as the positive one?
     
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