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Math Help - Showing a limit does not exist?

  1. #1
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    Showing a limit does not exist?

    \lim_{x \to 2 } \frac{3}{(x - 2)}

    I cheated by looking at a graph and it goes to \pm \infty, but when I try to solve it algbraically by substitution, I plug 2 in and get undefined in the denominator, as I'd expect with an infinite limit. But how do I show the algebraic work that the behavior of the function as the \lim_{x \to 2 } goes to both \pm \infty?
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  2. #2
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    Re: Showing a limit does not exist?

    Quote Originally Posted by LiberalArtMajorTakingCalc View Post
    \lim_{x \to 2 } \frac{3}{(x - 2)}

    I cheated by looking at a graph and it goes to \pm \infty, but when I try to solve it algbraically by substitution, I plug 2 in and get undefined in the denominator, as I'd expect with an infinite limit. But how do I show the algebraic work that the behavior of the function as the \lim_{x \to 2 } goes to both \pm \infty?
    Let f(x)=\frac{3}{(x - 2)} define x_n=2+\frac{1}{n}~~y_n=2-\frac{1}{n}

    It is clear that x_n\to 2^+~\&~y_n\to 2^-, but f(x_n)=3n\to\infty~\&~f(y_n)=-3n\to-\infty.
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    Re: Showing a limit does not exist?

    Where did 2+\frac{1}{n} and 2-\frac{1}{n} come from?
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