Proving Limit Exists: xy/(x^2+y^2)^.5

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Homework Statement

Prove or disprove that the following limit exists

lim x->0 and y->0 of
xy/(x^2+y^2)^.5

Homework Equations

0 < ((x-a)^2+(y-b)^2)^.5 < delta

|f(x,y)-L| < epsilon

The Attempt at a Solution

Taking the limit from both the x-axis and y-axis separately, L=0. So

0 < (x^2+y^2) < delta and |xy/(x^2+y^2)| < epsilon

We know that

|xy/(x^2+y^2)| = |x||y|/(x^2+y^2)

and
|y| = (y^2)^.5 <= (x^2+y^2)^.5

|x||y|/(x^2+y^2) <= |x|(x^2+y^2)^.5/(x^2+y^2)^.5 = |x|

|x| <= (x^2+y^2)<delta < epsilon

so delta = epsilon

Is this right?

 

[tiny-tim]

hi hover!

(have a square-root: √ and a delta: δ and an epsilon: ε and try using the X² tag just above the Reply box )

your method is basically correct, but your δ has to be the size of (x,y), not of x …

in other words, you can use the same method, to prove that for any ε, if |(x,y)| < ε, then |xy/√(x² + y²)| < ε

(but if you know polar coordinates, then use them instead, it's much easier! )

 

[hover]

Thanks!