Using the Delta-epsilon definition but for two variables?

[pr0me7heu2]

I have probably over thought the whole thing... but I can't seem to find any place to start with this one:

Using the formal definition of a limit:

f(x,y)= y / (x^2 + 1) e=0.05

 

[D H]

For multiple dimensions you need to extend the epsilon-delta notation to incorporate the epsilon-neighborhood of the point p₀ = (x,y), which is the set of all points p such that the distance between p₀ and p is less than epsilon.

 

[Kummer]

I have probably over thought the whole thing... but I can't seem to find any place to start with this one:

Using the formal definition of a limit:

f(x,y)= y / (x^2 + 1) e=0.05

The limit seems to be 0.
So if 0<|y|<\delta and 0<|x|<\delta.

\left| \frac{y}{x^2+1} \right| \leq \frac{|y|}{x^2} < \frac{\delta}{x^2}.

Now use single variable analysis that \frac{\delta}{x^2} can be made sufficiently small.

 

[HallsofIvy]

I have probably over thought the whole thing... but I can't seem to find any place to start with this one:

Using the formal definition of a limit:

f(x,y)= y / (x^2 + 1) e=0.05

limit as (x,y) goes to what?