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Find values for which the limit exists...

by twoflower
Tags: exists, limit, values
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twoflower
#1
Oct16-05, 07:50 AM
P: 368
Hi,
I'm given this problem:

Find conditions for variables a, b, c so that the limit

[tex]
\lim_{[x,y] \rightarrow [0,0]} \frac{xy}{ax^2 + bxy + cy^2}
[/tex]

exists.

What I have only found so far is that for all variables non-zero the limit doesn't exist. Anyway, I have no clue how to find the conditions for which it does. I tried a = b = c = 0, but it doesn't seem to help to me...

Thank you for the enlightenment.
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Lisa!
#2
Oct16-05, 08:35 AM
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OOps I was too late in deleting my post. Actually I made a mistake in solving that and yes, what makes me unsure is that I don't think we could use hopital rule for these kind of limit.
Lisa!
#3
Oct16-05, 08:42 AM
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You'd better to ask another homework helper, but I solve it in another way now . If you look at numerator nad denominator, you see both of them have xy. So?

twoflower
#4
Oct16-05, 02:45 PM
P: 368
Find values for which the limit exists...

Quote Quote by Lisa!
You'd better to ask another homework helper, but I solve it in another way now . If you look at numerator nad denominator, you see both of them have xy. So?
Ok, now it seems to me that the condition for the limit to exist is that a = c = 0.

Anyway, it is just the result of guessing method, is there any more exact approach to solve this?
HallsofIvy
#5
Oct16-05, 03:13 PM
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For functions like this, where you have two variables, I find it best to convert to polar coordinates. That way, exactly one variable, r, measures the distance to (0,0) which is the crucial factor. In polar coordinates,
[itex]x= r cos(\theta)[/itex] and [itex]y= r sin(\theta)[/itex] so
[itex]xy= r^2 cos(\theta)sin(\theta)[/itex], [tex]x2= r^2 cos^2(\theta)[/itex], and [tex]y^2= r^2 sin^2(\theta)[/itex].
Of course, then [itex]ax^2+ bxy+ cy^2= ar^2cos^2(\theta)+ br^2sin(
theta)cos(\theta)+ cr^2sin^2(\theta)[/itex] so that
[itex]ax^2+ bxy+ cy^2= r^2(acos^2(\theta)+ bsin(
theta)cos(\theta)+ csin^2(\theta)[/itex].

That means that
[tex]\frac{xy}{ax^2+ bxy+ cy^2}= \frac{sin(\theta)cos(\theta)}{acos^2(\theta)+ bsin(\theta)cos(\theta)+ csin^2(\theta)}[/tex].

Notice that there is no "r" in that! This can have a limit as r-> 0 only if it does NOT depend on [itex]\theta[/itex]- it is a constant. One obvious choice for a,b,c is a= c= 0, b= 1 but there may be others.


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