A few questions regarding mutivariable limits. I know that if we take the limit of a multivariable function along different paths and we arrive at different limits then the limit does not exists.
However would it be correct to say that;
If we take the limit along a path and the limit exists yet if 
we take the limit along another path and the limit does not exist then 
the limit does not exist??
This screams TRUE, but I like to tread carefully.
also, say if we would like to find the limit of a function at some point $(a,b)$ could I let $x=a$ and $y = b+n$ and see what happens as $n \rightarrow 0$ then $x=a+n$ and $y = b$ as $n \rightarrow 0$ then if we get different limits the limit at $(a,b)$ does not exist?
 A: What do you mean by "if we take the limit along another path and the limit does not exist"? If this means that when $(x,y)\to(a,b)$ along some path, we get $\lim f(x,y)=\pm\infty$, and along some other path we get
$$\lim f(x,y)=L$$
then the limit does not exist. You are correct in all your other statements. 
The $\epsilon-\delta$ definition of a multivariable limit of a function $f(\mathbf{x})$ in $\mathbb{R}^n$ is that the limit $\lim_{\mathbf{x}\to\mathbf{a}} f(\mathbf{x})=L$ exists if and only if for all $\epsilon>0$ there exists a $\delta>0$ such that
$$|f(\mathbf{x})-L|<\epsilon$$
whenever
$$0<|\mathbf{x}-\mathbf{a}|<\delta$$
where $\mathbf{x}=(x_1,x_2,\ldots,x_n)$ and $\mathbf{a}=(a_1,a_2,\ldots,a_n)$ are points in $\mathbb{R}^n$, and
$$|\mathbf{x}-\mathbf{a}|=\sqrt{(x_1-a_1)^2+(x_2-a_2)^2+\ldots+(x_n-a_n)^2}$$
is the distance between the two points.
For $\mathbb{R}^2$ and $\mathbb{R}^3$, this corresponds to the function getting arbitrarily close to $L$ whenever we restrict the domain to a circle or sphere around the point $\mathbf{a}$.
