So I'm given a function and I have to know what the limit is without drawing a graph. I've learnt that if the denominator is 0 when you sub the number into x, then you have to factorise the function. and if you can't, then the limit does not exist. But then I came across this question: lim x->-2 [[x]].
[[x]] means the greatest integer function. so in this case x should equal -2? I'm not quite sure how to do this question, but the answer is 'limit does not exist'.

This one may be hard to do without knowing what the greatest integer function means, or seeing the graph visually. A limit only exists when it approaches from both the positive and negative - try to think of this question that way.

I read on the greatest integer function earlier and I think it just means to round down to the smaller integer value.
so [[5.9]]=5 and [[-2.1]]=-3.
We're expected to solve these limits algebraically, so graphing wouldn't be the right method.
Thanks.

Yup, you're right about the GI function. The function brings x to the nearest integer less than x. So, as I hinted at, try to evaluate the limit: lim x->-2(-) and lim x->-2(+), aka: evaluate from the left and the right. If the limit exists, it must be approached from both sides.

Edit: Even though the solution requires an algebraic solution, looking at the graph may help you realise how to do so.