Can you use the squeeze rule for the following: rather than multiplying by the conjugate?

I did the following: . So wouldn't the limit be ?

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- October 18th 2007, 10:13 AMshilz222Squeeze Rule? R2 limit
Can you use the squeeze rule for the following: rather than multiplying by the conjugate?

I did the following: . So wouldn't the limit be ? - October 18th 2007, 10:39 AMPlato
- October 18th 2007, 01:47 PMshilz222
What a dumb mistake. Aargh.

- October 18th 2007, 01:57 PMPlato
Can you use polar substitution?

.

You will get a function of alone and then let - October 18th 2007, 02:11 PMshilz222
yes it would be so now you can use L'Hopitals rule.

- October 18th 2007, 02:16 PMPlato
Very good. Is the limit 2?

- October 18th 2007, 02:21 PMshilz222
yes. the limit is 2. I made a mistake on that one on a test.

- October 18th 2007, 02:38 PMThePerfectHacker
Just rationalize!

- October 18th 2007, 02:41 PMshilz222
Yes I missed that. I knew I should have done that. But alas, I was under a time constraint.

- October 20th 2007, 11:33 PMshilz222
Can you say the limit is 0 if we restrict ? Or is this not allowed?

- October 20th 2007, 11:39 PMJhevon
- October 21st 2007, 12:28 AMshilz222
But we have Real Analysis and we have Complex Analysis. So why cant we consider the limit in ? (i.e. Integer Analysis)? Couldn't we also consider this same limit in ? It wouldnt make sense in but would it make sense in ?

- October 21st 2007, 12:31 AMJhevon
and are complete sets (the completeness axiom applies). this is a property that makes it possible to evaluate limits, because we need to make things as close as possible. this is not possible with the integers, since there are gaps 1 unit in length between integers. finding limits under those conditions make no sense really, since we can't make things arbitrarily close (recall the epsilon-delta definition of the limit of a function)

- October 21st 2007, 12:35 AMshilz222
But the definition of the limit (epsilon-delta) is based on ? So we cant really use a definition of a limit that is defined in in right? So basically when you take a limit in the closest you can be is 1 unit away. So cant you restrict the definition of a limit? You are still taking a limit right?

Actually I think you can. by the Dedekind cuts. So the definition that applies in will also apply to but not vice-versa. - October 21st 2007, 12:49 AMJhevon
we can take limits in as well. even if you could come up with a construction for limits in the integers, it would be essentially meaningless. since it would give rise to things like continuity at a point where there is actually a gap. it would simply make no sense, and won't be very interesting. in any case, if you considered only integers here, the answer would still be 2, so why even bother with it?