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# limits

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1. limits
Ok, here's an interesting limit question. I thought that when you have a radical, you multiply top and bottom by the conjugate,true? How about this one?

lim sqrt(9+x^2)/(x-3)
x-->3

So what happens here? do you square top and bottom to get rid of the radical or???
2. Re: limits
I don't know what theorems you've been taught but putting mathematical rigour aside for a moment, this limit is clearly non-existent. When you approach 3 from above you get a different answer from when you approach it from below. You could try bounding the function by some other (easier to work with) function to prove this.
Last edited by electriic_ink; 02-04-2012 at 17:46.
3. Re: limits
(Original post by Snazarina)
Ok, here's an interesting limit question. I thought that when you have a radical, you multiply top and bottom by the conjugate,true? How about this one?

lim sqrt(9+x^2)/(x-3)
x-->3

So what happens here? do you square top and bottom to get rid of the radical or???
The limit doesn't exist. (Suppose it did, and the limit = L, then (as long as the limits in this equation exist, which they clearly do).

Since the RHS = 0 and the LHS = 3 sqrt 2, this is not possible).
4. Re: limits
You're welcome.
5. Re: limits
True - no limit exists.
You can also see this (assuming you want to) by using a combination of limit rules: Limit[Sqrt(something)] = Sqrt[Limit(that something)], together with LHops rule to read:

Sqrt[Limitx->3{LHops rule for 9+x^2/(x-3)^2}]= undefined - with the assumption that x>0 w.l.o.g
Last edited by Hasufel; 04-04-2012 at 16:25.
6. Re: limits
(Original post by Hasufel)
True - no limit exists.
You can also see this (assuming you want to) by using a combination of limit rules: Limit[Sqrt(something)] = Sqrt[Limit(that something)], together with LHops rule to read:

Sqrt[Limitx->3{LHops rule for 9+x^2/(x-3)^2}]= undefined - with the assumption that x>0 w.l.o.g
You can't apply L'Hopital unless the original limit is indeterminate.

(e.g. applying L'Hopital to (9+(x-3))/(x-3) at x = 3 would tell you the limit = 1, which it isn't...)
7. Re: limits
Oh gee thanks everyone. So it's simply non existent. And how do u tell if it's existent or not? I mean they all look the same
8. Re: limits
(Original post by Snazarina)
Oh gee thanks everyone. So it's simply non existent. And how do u tell if it's existent or not? I mean they all look the same
If you have one variable then you just need to show that the right hand limit and the left hand limit approach the same number.
9. Re: limits
(Original post by DFranklin)
You can't apply L'Hopital unless the original limit is indeterminate.

(e.g. applying L'Hopital to (9+(x-3))/(x-3) at x = 3 would tell you the limit = 1, which it isn't...)
WRONG! - 2/3rds - but i take your point
Last edited by Hasufel; 08-04-2012 at 16:49.
10. Re: limits
(Original post by Hasufel)
WRONG!
Wut? L'Hôpital doesn't apply here. Just because it gives you the right answer doesn't mean the method is right.
11. Re: limits
(Original post by DFranklin)
You can't apply L'Hopital unless the original limit is indeterminate.

(e.g. applying L'Hopital to (9+(x-3))/(x-3) at x = 3 would tell you the limit = 1, which it isn't...)
True! i bow to thee! (silly me!)

EDIT: apologies! wasnt concentrating (my brain is fried - im studying from different books at once, you know: arse....elbow...)
Last edited by Hasufel; 08-04-2012 at 16:52.

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Last updated: April 8, 2012
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