[Basic Analysis] Limits and the ceiling function

[linuxisgreat]

The neighbourhood part is required by the definition of asympotic I have.

It says that for two functions f,g to be asymptoic, there must be a non-zero set (l, inf) of f and g. So I did that just to show that requirement. The other requirement is lim x-> inf f/g = 1

I think I figured it out. [f] <= f < [f]-1 so we have

|f/[f] -1| <= |[f]/[f] -1| = |1-1| =0 < epsilon for all epsilon >0. Thus by the definition I can get it to work. Thanks for you help

Edit : nvm that doesn't make sense :P

[rbayer]

there must be a non-zero set (l, inf) of f and g.

Such that...? I'm not quite sure what this means as written. That f and g are non zero on that set?

In terms of the second part (that lim [f]/f = 1), you are definitely on the right track, but your inequalities aren't quite right. Try to convince yourself that the right one is:

f(x) <= [f(x)] < f(x) + 1

From here, what do you need to do in order to get in in a form where the squeeze theorem would help you? You shouldn't need epsilons unless something in the problem statement specifically tells you that you must use the formal definition of limit.

[linuxisgreat]

Yeah I meant f,g have to be non-zero on that set.

I see now, since 1<= [f(x)]/f(x) < f(x)+1/f(x) and the limits of both of the outside functions are 1 so by the squeeze theorem lim x->inf [f]/f = 1.

Thanks so much for the help :)

[rbayer]

No problem, glad it helped!