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Math Help - function doesn't tend to -infty or infty, how to prove it diverges?

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    Junior Member mremwo's Avatar
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    function doesn't tend to -infty or infty, how to prove it diverges?

    How would I show that lim as x goes to 1 of x/(x-1) does not exist, using a rigorous proof?

    I am a little confused since this function neither tends to -infinity or infinity (at least not that I think it does). I started off by proving this fact, but it turns out that I don't think it is getting me anywhere. Any suggestions?

    Thanks.
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  2. #2
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    Might help if you write the function as 1 + 1/(x-1) and go from there.
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    Quote Originally Posted by mremwo View Post
    How would I show that lim as x goes to 1 of x/(x-1) does not exist, using a rigorous proof?

    I am a little confused since this function neither tends to -infinity or infinity (at least not that I think it does). I started off by proving this fact, but it turns out that I don't think it is getting me anywhere. Any suggestions?

    Thanks.
    It tends to +infinity as x goes to 1 from above, -infinity as x goes to 1 from below. Either is enough to show that it does not converge. Remember that, to prove a general statement is not true, a counter example is sufficient. Take \epsilon= 1, for simplicity. If x> 1 then let \delta= x- 1> 0. x= 1+ \delta so \left|\frac{x}{x- 1}- 1\right|= \left|\frac{1+ \delta}{\delta}- 1\right|= \frac{1}{\delta} (notice how debsta's "1+ 1/(x-1)" came into that) which will be larger that \epsilon= 1 as long as [itex]\delta< 1[/tex]. We cannot make it small by taking \delta smaller.
    Last edited by HallsofIvy; March 1st 2011 at 03:27 AM.
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