How does the following notation read in plain English:
$$\lim\limits_{x \to \infty \\ y \to \infty} F(x, y) = 1$$
As far as I can read:
- If $x$ and $y$ approaches $\infty$, the value of function $F(x, y)$ approaches $1$.
i.e.- The value of the function $F(x, y)$ is $1$ somewhere near but before $(x, y) = (\infty, \infty)$.
- The $limit$ of the function $F(x, y)$ is $1$ at $(x, y) = (\infty, \infty)$.
On
By the following
$$\lim\limits_{x \to \infty \\ y \to \infty} F(x, y) = 1$$
it seems you are referring to a case where both $x$ and $y$ tends to positive infinity as for example
$$F(x,y)=1+\frac1x+e^{-y}$$
and the first sentence is preferable to decribe that situation "when both $x$ and $y$ approch $\infty$ then $F(x,y)$ approches $1$".
Note that in the standard definitin of limit at $\infty$ we are requiring that $\|(x,y)\| \to \infty$ and in that case we can say that the "limit of $F$ is $1$ as $x^2+y^2 \to \infty$" that is
$$\lim\limits_{x^2+y^2 \to \infty} F(x, y) = 1$$
(1) is correct. (2) and (3) are not.
What you write as "$(\infty, \infty)$" makes no sense. Infinity is not a number, and there is no place with those coordinates, so it makes no sense to ask for the value of $F$ there.
Informally: the values of the function $F$ are as near to $1$ as you please everywhere outside a circle of large enough radius. The value may never actually be $1$. Think about the function given by $$ F(x,y) = 1 - \frac{1}{x^2 + y^2 + 1} . $$
Edit. There is in fact some ambiguity about whether we are looking at the two variables $x$ and $y$ as independently large and positive, or whether the OP intends to ask about what happens when $||(x,y)||$ is large. In either case (2) and (3) make no sense. Assertion (1) will be correct if the ambiguity is resolved by the context.