Evaluate the limit.

We use this exercise (Section 7.11, Exercise #4) to evaluate,

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Stumbling Robot

A Fraction of a Dot
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Find the limit as *x* goes to 1 of *(1 / log x – 1 / (x-1))*

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I have worked through exercises nr. 6 – 29 and It actually isn’t that complicated. I have one thing however that confuses me and caused some problems (especially at ex. nr. 20 -22). How do I know to what degree I have to take the polynomial? The choices in the solutions sometimes seemed arbitrary to me. I couldn’t see a pattern there. Because of that I sometimes chose to many degrees which resulted in an algebraic mess and sometimes to few degrees which when I evaluated the expression led me back to 0/0 or similiar. Is that something that comes with experience or is there a fixed set or rules that tells me to what degree I have to take the polynomial in each case?