Find limit, if it exists, of xy/sqrt(x^2+y^2) as (x,y)->(0,0

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MarkSA

Junior Member
Joined
Sep 8, 2007
Messages
243
Hello,

Q: Find the limit, if it exists... If it does not exist, show that it does not exist.
lim (x,y)->(0,0) of: (x*y)/sqrt(x^2 + y^2)

I am having real trouble with this. To show that the limit does not exist, I have to show that the limit to (0,0) is different for at least two different paths.
Here is what I have tried:
Let x=0 (.. limit = 0)
let y=0 (.. limit = 0)
let y=mx (.. limit = 0)
let y=x^2 (.. limit = 0)

So at this point i'm starting to think the limit really is zero. But how do I prove it exactly? ie how can I say definitively that this limit is zero regardless of path.
 
Re: Find the limit, if it exists...

Along the paths x=y and x=-y, both of which approach (0,0), are the limits the same?
If not, then what does that tell you?
 
Re: Find the limit, if it exists...

Wait.. isn't the limit zero in both cases? But does that prove anything? All of my other limits were zero too.
 
Re: Find the limit, if it exists...

Anyone know? This is one i'd really like to figure out before the test
 
Re: Find the limit, if it exists...

MarkSA said:
Hello,

Q: Find the limit, if it exists... If it does not exist, show that it does not exist.
lim (x,y)->(0,0) of: (x*y)/sqrt(x^2 + y^2)

I am having real trouble with this. To show that the limit does not exist, I have to show that the limit to (0,0) is different for at least two different paths.
Here is what I have tried:
Let x=0 (.. limit = 0)
let y=0 (.. limit = 0)
let y=mx (.. limit = 0)
let y=x^2 (.. limit = 0)

So at this point i'm starting to think the limit really is zero. But how do I prove it exactly? ie how can I say definitively that this limit is zero regardless of path.
Click to expand...

Try it in polar coordinates. Assume you approach along some path theta = t.

xy
-------------
sqrt(x^2 + y^2)

Take x = r cos t, y = r sin t, for some fixed t.

r^2 cos t sin t
---------------
r
= r cos t sin t

But cos t sin t is bounded.

Does that do it?
 
varmanamburi said:
The function f(x, y) = xy/sqrt(x^2 +y^2) is undefined at (x, y) =(0, 0). if we let (x, y) approach (0, 0) along the x-axis (y=0) then f(x, y) = f(x, 0) =0. Similarly at all points along y-axis, we have f(x, y) = f(0, y)=0. However, at points of the line y=kx, f(x, y) = f(x, kx) = kx^2/sqrt(x^2 +k^2x^2) = x/sqrt(1+k^2). Thus, the limit is 0 along x- and y-axes and along the line x=y when (x, y) approaches to (0, 0). Along the level curve y = x^2, f(x, y) = x^3/sqrt(x^2 +x^4) = x^2/sqrt(1+x^2). It approaches to 0 as (x, y) approaches to (0, 0). It is clear that the limit exists. (www.math-and-stat.com)
Click to expand...

It can be seen in the graph:
ttt.jpg
 
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