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I just started studying calculus for next year and I'm stumped on two step-wise probs

For what values of a does lim x->a [[x/2]] exist?

and

f(x)= sin pi/[[x]] a=3
FIND
a)lim x->a- f(x)
b)lim x->a+ f(x)
c)lim x->a f(x)

I'm no good at these types and help would be greatly appreciated.

By D00ditzme (D00ditzme) on Thursday, May 20, 2004 - 09:32 am: Edit

bump

By Edward (Edward) on Thursday, May 20, 2004 - 11:20 am: Edit

For the first one, I'm not too sure about it, but I think it is all real numbers except even numbers. If you try putting an even number in for 'a', the left and right hand limits for the greatest integer function are different, meaning that lim x->a does not exist.

For the second one:
a) lim x->3- f(x) will become sin pi/2. The [[3]] becomes 2 because if you look examine values between 2 and 3 (for [[a]]), you'll see that they all equal 2. For example, [[2.1]]=2, [[2.2]]=2, [[2.3]]=2. Thus, the limit as x approaches 3 from the left is 2 (for lim x->3- [[3]]).
Therefore, sin pi/2 = 1.

b) In this case, we have to see what the values of the function will be for values of 'a' greater than 3 (because x is approaching a from the right). [[3.3]]=3, [[3.2]]=3, [[3.1]]=3. Thus, the limit for the greatest integer part of the function will reduce down to 3.
Therefore, f(x)=sin pi/3, which is root 3 over 2.

c) From parts a and b, we saw that the left and right hand limits are different. Therefore, the limit for the function as x approaches 3 does not exist.

I hope this is clear. Good luck with your calculus!

By D00ditzme (D00ditzme) on Thursday, May 20, 2004 - 12:17 pm: Edit

thanks a lot, it's a bit more clear now