Solved: Two Variable Limits Continuity at (0,0)

[misterau]

Homework Statement

The real valued function f of two variables is defined by f(x,y) =
tan( (1/2) *(pi) *sqrt( (x^2 + y^2) )/( sqrt( (x^2 + y^2) ) for each (x,y) satisfying 0 < x^2 + y^2 < 1.

How should f(0,0) defined so that f is continuous at (0,0)?

Homework Equations

The Attempt at a Solution

I wasn't sure whether the question was asking show the limit exists or doesn't exist? I think asking me to show it exists.
I tried using polar co ordinates
y=r*sin\theta , x=r*cos\theta
=tan( (1/2) * (pi) * r) / r
= 1/cos( (1/2) * (pi) * r ) * sin((1/2) * (pi) * r)/r
= 1 * sin((1/2) * (pi) * r)/r

I got stuck here was trying to use the identity lim->0 sinx/x = 1
I know you can use squeeze theorem to prove limits exist, but I don't really understand how it works for two variables.
In one variable limits if 0 < |x-a|<\delta then |f(x)-L|< \epsilon
but in two variable limits what is |x-a|?

 

[HallsofIvy]

since sin(x)/x-> 1, for sin(\pi r/2)/r, letx= (\pi/2)r. What is r equal to in terms of x? Replace the r in the denominator by that.