How Do I Construct an Epsilon-Delta Proof for a Function Mapping R2 to R1?

[Lancelot59]

I have a problem on a take-home test, so I can't ask about the specific problem. So this is just going to be a general, how do I put stuff together problem.

I have a function of x and y that maps R² into R¹. The limit as (x,y)->(0,0) is zero, and I've worked through the various paths already.

What I have so far for the proof is this:

Let $$\epsilon >0 \newline$$

$$|\vec{F}(x)-\vec{L}|<\epsilon$$ whenever $$0<||\vec{x}-\vec{a}||<\delta$$

So the way I see this as delta gets closer to zero, the between the inputs and the limit inputs should be greater than zero, but less than a certain difference.

So if I put that last bit together:
$$0<||(x,y)-(0,0)||<\delta \rightarrow 0<\sqrt{x^{2}+y^{2}}<\delta$$

So now what do I do?

 

[lippyka]

Is this enough of a proof?No, this is not a complete proof. To finish the proof, you need to show that for any given ε > 0, there exists a δ > 0 such that |\vec{F}(x)-\vec{L}|<\epsilon whenever 0<||\vec{x}-\vec{a}||<\delta. You can do this by finding an expression for \delta in terms of \epsilon, and then showing that it works for all \epsilon > 0. For example, you could say that \delta = \sqrt{\epsilon}\, and then show that if 0<||(x,y)-(0,0)||<\sqrt{\epsilon} then |\vec{F}(x)-\vec{L}|<\epsilon.