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Math Help - limits

  1. #1
    benji
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    Angry limits

    Why is the limit of (x-1)/(x-3) does not exist and the limit of x-1/(x-3)ex2 for both as x approches 3 from the left and the right
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  2. #2
    MHF Contributor red_dog's Avatar
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    \displaystyle\lim_{x\nearrow 3}\frac{x-1}{x-3}=\frac{2}{-0}=-\infty

    \displaystyle\lim_{x\searrow 3}\frac{x-1}{x-3}=\frac{2}{+0}=\infty

    So, the one-sided limits are different and the function has no limit for x=3

    The second one I think is \displaystyle\lim_{x\to 3}\frac{x-1}{(x-3)^2}.

    In this case, \displaystyle\lim_{x\nearrow 3}\frac{x-1}{(x-3)^2}=\frac{2}{+0}=\infty

    and \displaystyle\lim_{x\searrow 3}\frac{x-1}{(x-3)^2}=\frac{2}{+0}=\infty

    So, the one-sided limits are equal and \displaystyle\lim_{x\to 3}\frac{x-1}{(x-3)^2}=\infty.
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