I remember when I was doing the HSC that I thought you just had to divide by the highest power of x... and that seemed to work for most cases (possibly because, by chance or otherwise, they always put the larger polynomial in the denominator instead of the numerator).
If you look at the example I did, you'll get y -> 0 as x -> oo if you try to divide by x instead of just x.
Dividing by the highest power in the denominator was something I picked up at uni.
Hmm, you should observe that, by factorising, you can rewrite the fractional function as the result you get when dividing by x (which is essentially what you're doing anyway).
....lim................x..
x -> oo..........x + 4
----------->........x......
..................x(1 + 4/x)
----------->......x......
..................1 + 4/x