epsilon delta criterion for continuous functions

Hi there!

Iīm given a rational function and fixed epsilon and x_0.

How do I find a delta to this epsilon in general?

Hereīs what I do:

I use |f(x)-f(x_0)|<epsilon and try to do some algebra till I get it in the form: |x-x_0|.A<epsilon

unfortunately A is in most if the cases dependent on x.

So how can I practically get rid of the x in A so that I can divide by A and set epsilon/A=delta ?

hereīs and example: f(x)=1/x

|1/x-1/x_0|<e

|x-x_0|/|xx_0|<e

what do I do now? I tried using inequalities but it went all wrong, since I cannot use >=, cause I could thereby possibly get something greater than e, all thatīs left to me is using <= or..?

thanks for the help in advance!