Thread: 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.

Might help if you write the function as 1 + 1/(x-1) and go from there.

Originally Posted by mremwo

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 , for simplicity. If x> 1 then let . so (notice how debsta's "1+ 1/(x-1)" came into that) which will be larger that as long as [itex]\delta< 1[/tex]. We cannot make it small by taking smaller.