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Math Help - Limits

  1. #1
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    Limits

    Let f(x) = |x|^{-1/2} for x\neq0. I need to show that \lim_{x\to0+} f(x)= \lim_{x\to0-} f(x)=+\infty...
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  2. #2
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    Hello, let me try to help you

    1) Try when x<0, then |x|=-x.
    2) Try when x>0, then |x|=x.

    Take the limit, and tell what you get.

    Hugs
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  3. #3
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    Quote Originally Posted by CrazyCat87 View Post
    Let f(x) = |x|^{-1/2} for x\neq0. I need to show that \lim_{x\to0+} f(x)= \lim_{x\to0-} f(x)=+\infty...
    To prove that:

    lim_{x\to 0^+} f(x) =+\infty you must prove that:

    for all ε>0 there exists a δ>0 such that :

    for all ,x : if 0<x<δ , then f(x)>ε
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  4. #4
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    Quote Originally Posted by miguemate View Post
    Hello, let me try to help you

    1) Try when x<0, then |x|=-x.
    2) Try when x>0, then |x|=x.

    Take the limit, and tell what you get.

    Hugs
    And, therefore, it is sufficient to prove that \lim_{x\to +\infty}|x|^{1/2}= +\infty.
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  5. #5
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    Quote Originally Posted by HallsofIvy View Post
    And, therefore, it is sufficient to prove that \lim_{x\to +\infty}|x|^{1/2}= +\infty.
    Why would that be sufficient?
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  6. #6
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    Because making the substitution y= -x in \lim_{x\to -\infty} |x|^{-1/2} gives \lim_{y\to\infty} |-y|^{-1/2}= \lim_{y\to\infty} |y|^{-1/2} which is, of course, the same as \lim_{x\to \infty}|x|^{-1/2}
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  7. #7
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    ok I'm done proving it, thanks everyone for all your help!!
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