# Thread: Multivariate limit of a square root function

1. ## Multivariate limit of a square root function

what is the limit of Sqrt[x^2 + y^2] as (x,y) approach (0,0)?

I wrote 0 on the exam but I realized it probably doesn't exist because the sqrt fcn is defined only for positive values of x and y, what do you guys think?

2. ## Re: Multivariate limit of a square root function

I think $x^2 + y^2 \ge 0$ for all $x$ and $y$.

Writing $x=r\cos\theta$ and $y=r\sin\theta$ should give you a good insight into the problem.

3. ## Re: Multivariate limit of a square root function

What you wrote on the exam is correct. If you look at the definition of a limit, it requires that "... for all $\displaystyle (x,y)$ in the domain of F with $\displaystyle 0<|(x,y)-(a,b)|<\delta$, $\displaystyle |F(x,y)-L|<\epsilon$."

So you don't need an $\displaystyle (x,y)$ that makes $\displaystyle F(x,y)$ negative, you just need to look to see what F does to $\displaystyle (x,y)$ pairs near the origin.

- Hollywood