# Two variable limits

1. Mar 31, 2009

### misterau

1. The problem statement, all variables and given/known data

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)?

2. Relevant equations

3. 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|?

2. Apr 1, 2009

### HallsofIvy

Staff Emeritus
since sin(x)/x-> 1, for $sin(\pi r/2)/r, let$x= (\pi/2)r[itex]. What is r equal to in terms of x? Replace the r in the denominator by that.