# Thread: Epsilon Delta Proof of a Limit

1. ## Epsilon Delta Proof of a Limit

The problem is give a rigerous epsilon delta proof that lim (x,y) (0,0) (x^3 + y^3)/(x^2 + y^2)

Heres what I have. Given any ε > 0, we can find a corresponding δ > 0 such that if 0 < || (x,y) - (0,0)|| < δ , then (x^3 + y^3)/(x^2 + y^2) < ε

0 < √(x^2 +y^2) < δ

√(x^2) =|x|< δ
√(y^2) = |y| < δ

I'm stuck after this part. Is the above part even correct? If so, how do I finish i then?

Thanks

2. Originally Posted by meks08999
The problem is give a rigerous epsilon delta proof that lim (x,y) (0,0) (x^3 + y^3)/(x^2 + y^2)

Heres what I have. Given any ε > 0, we can find a corresponding δ > 0 such that if 0 < || (x,y) - (0,0)|| < δ , then (x^3 + y^3)/(x^2 + y^2) < ε

0 < √(x^2 +y^2) < δ

√(x^2) =|x|< δ
√(y^2) = |y| < δ

I'm stuck after this part. Is the above part even correct? If so, how do I finish i then?

Thanks
$\displaystyle \left|\frac{x^3+y^3}{x^2+y^2} \right| =\frac{|x^3+y^3|}{x^2+y^2} \leq \frac{x^2|x| + y^2|y|}{x^2+y^2} \leq |x| + |y| < 2 \delta.$ so we need to choose $\displaystyle \delta = \frac{\epsilon}{2}.$

3. Thanks for the help.