Hiya, I'm having to learn about limits really quickly and am struggling to fully grasp the concept. While I understand the basic concept, I'm having trouble figuring out what can be considered as (dis-)proof of a limit, and also the Epsilon-Delta definition of a limit. I can't seem to find anything that offers an explaination of these which is not in mathematical terms - I am not a mathematician, hence the difficulty.

I have a few example functions and would appreciate any help, although I would prefer more descriptive help rather than just the maths. All functions are defined as 0 for (x,y)=(0,0) and I would like to test the continuity at (0,0).

1. $\displaystyle f(x,y)=\frac{xy^3}{x^2+y^6}$

2. $\displaystyle f(x,y)=\frac{xy}{x^2+xy+y^2}$

3. $\displaystyle f(x,y)=\frac{x^2+y^2}{x^2-y^2}$

Here are my approaches to these functions...

Function 1.

We can rewrite the function as $\displaystyle f(x,y)=\frac{\sqrt{x^2}}{\sqrt{x^2+y^6}}*\frac{\sq rt{y^6}}{\sqrt{x^2+y^6}}$. My notes feature fractions with the same powers for both numerator and denominator - is this proof of a limit?

Function 2.

If we consider the result of $\displaystyle f(x,y)$ along the line $\displaystyle f(x)=y$, the function becomes $\displaystyle f(x,y)=\frac{x^2}{x^2+x^2+x^2} = \frac{1}{3}$. As this is constant, but the function is defined as (0,0) at (x,y)=(0,0), we have a large step rather than a gradual convergence, therefore $\displaystyle f(x,y)$ is not continuous at (0,0).

Function 3.

If we consider the function when $\displaystyle x=0$ and $\displaystyle y=1$, the result of the function, the result will be -1 and 1 respectively. As with function 2, we have a constant value along these lines everywhere except (0,0), therefore $\displaystyle f(x,y)$ is not continuous at (0,0).

Thank you in advance for any help or advice you can offer.