Prove by Beta and Sigma limit definition

[kidia]

Please any help.

Use the symbols $$\beta$$ and $$\sigma$$ definition of limit to prove that limit (x,y)$$\Longrightarrow$$(0,0)x+y/x$$\x^2$$+y$$\y^2$$=0

 

[EnumaElish]

What is a beta and sigma limit definition? Same as the epsilon and delta definition? Here are some examples:
http://archives.math.utk.edu/visual.calculus/1/definition.6/
http://omega.albany.edu:8008/calc3/several-vars-dir/lim-epsilon-delta-m2h.html
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html

 

[rachmaninoff]

I think beta/sigma definitions are a 2D analogue to epsilon/deltas? I'm afraid I can't find any reference to them, you'll have to define them please.

 

[quasar987]

You want to show that for a given $\beta >0$, you can find a number $\sigma >0$ such that when the distance from the origin of the point (x,y) is smaller than $\sigma$, then this implies that

[tex]\frac{x+y}{x^2+y^2}<\beta[/itex]

So we kinda want to find a function $\sigma(\beta)$.

The statement "the distance from the origin of the point (x,y) is smaller than $\sigma$" is written mathematically as $\sqrt{x^2+y^2}<\sigma$

There are many solutions but here's a hint based on one:

Use the fact that $x+y \leq (x^2+y^2)^2$ coupled with the hypothesis $\sqrt{x^2+y^2}<\sigma$ to define a function $\sigma(\beta)$.