How to check if limit does not exist

Oct 2012
2
0
Perth
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'.
 

Kyo

Sep 2012
30
0
Canada
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.
 
Oct 2012
2
0
Perth
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.
 

Kyo

Sep 2012
30
0
Canada
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.
 
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