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Math Help - Limits!!!

  1. #1
    Newbie
    Joined
    Mar 2008
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    Limits!!!

    First post!!! I have two questions guys (and girls)

    Seeing as how these questions are so hard to write down in posts I have uploaded an image of them.

    FreeImageHosting.net Hosting Service

    1) Let the function f(x) be defined by

    f(x) = 1/6((sqrt(x) -1)/(sqrt(x^2-1))) + (sqrt(7x)+sqrt(7))/(3(sqrt(x+3))) + x/6(sqrt((x-1)/(x^2-1)))

    It can be shown that
    f(1.1) approximately equals 1.05891
    f(1.01) approximately equals 1.00982
    f(1.001) approximately equals 1.00205

    Find the exact value of f(x) as x approaches 1 from the right(1+) (so lim x->1+ f(x)
    The + is part of the 1,meaning as x approaches 1 from the right)

    My answer for this is (2sqrt(7)+sqrt(1/2))/6. Is this right? If not what is the right anwer and how to do it?

    2) Use the squeeze theorem to prove that the limit of

    x(sin^2(1/x)) = 0

    as x approaches 0

    Can somebody show me how to do this? I really need help on this one!!!

    Thanks in advance
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  2. #2
    Junior Member roy_zhang's Avatar
    Joined
    Mar 2008
    Posts
    64
    1) I think your answer is correct.

    2) Since 0\le\sin^2(\frac{1}{x})\le 1, you then have 0\le x\sin^2(\frac{1}{x})\le |x|
    Apply Squeeze Theorem from here.

    Roy
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