What did I do wrong?

Question: Evaluate limx->0 [(1)/(x-2)] / [x]
My work:
[(1/x-2) + (1/2) / (x)] multiply [(2(x-2)/1) / (2(x-2)/1)]
(2-3x) / (2x^2) = (2/0) = limit does not exist

But apparently the limit does exist, so what did I do wrong? I think I have to eliminate the x in the denominator but I am not sure how to do this.

Any help is appreciated.