Infinity is not a number.
So there there is no general meaning to any "infinity arithmetic" expression.
Sometimes, though, there is a limit theorem which can be interpreted as an infinity arithmetic expression.
Here's one example of such a theorem:
Theorem: Given sequences $(x_n)$ and $(y_n)$ in $\mathbb R$, if $\lim_{n \to \infty} x_n = \infty$, and if $\lim_{n \to \infty} y_n = \infty$, then $\lim_{n \to \infty} (x_n + y_n) = \infty$.
Because of this theorem, one might argue that it is fair to "split the limits", as you say, resulting in the "infinity arithmetic" expression.
$$\infty + \infty = \infty
$$
Fine so far.
But just because one can write an "infinity arithmetic" expression does not mean there is a theorem supporting that expression.
So, for example, there is NO theorem like this:
False Theorem: Given sequences $(x_n)$ and $(y_n)$ in $\mathbb R$, if $\lim_{n\to\infty} x_n = \infty$ and $\lim_{n \to \infty} y_n = \infty$ then $\lim_{n \to \infty} (x_n - y_n) = $ BLAH.
It doesn't matter what you substitute for BLAH, the resulting statement will be false. Whether you substitute BLAH $=0$, or BLAH $= 1$, or BLAH $=42$ or BLAH $=$anything else, the resulting statement will be false.
To prove this, let me give you two counterexamples:
- Counterexample 1: If $x_n = n$ and $y_n = n$ then $\lim_{n\to\infty} x_n = \infty$, and $\lim_{n \to \infty} y_n = \infty$, and
$$\lim_{n \to \infty} (x_n - y_n)= \lim_{n \to \infty} (n - n) = \lim_{n \to \infty} 0 = 0
$$
- Counterexample 2: If $x_n = n$ and $y_n = n-1$ then $\lim_{n \to \infty} x_n = \infty$ and $\lim_{n \to \infty} y_n$ and
$$\lim_{n \to \infty}(x_n-y_n) = \lim_{n \to \infty} (n - (n-1)) = \lim_{n \to \infty} 1 = 1
$$
So, if you tried to convince me that the "False Theorem" was true using the substitution BLAH $=0$, I would show you Counterexample 2. And if you tried to convince me that the "False Theorem" was true using any substitution not equal to $0$, such as BLAH $=1$ or BLAH $=42$ or BLAH $=\infty$ or BLAH $=$anything else not equal to zero, then I would show you Counterexample 1.
In Calculus 1 we teach that the expression "$\infty-\infty$" is an indeterminate form. What this really means is what I've said above: there is no limit theorem which justifies any evaluation of $\infty-\infty$. When you encounter what looks like an $\infty-\infty$ expression, your best mathematical strategy is to DO SOMETHING ELSE, i.e. to re-evaluate the expression, rewrite it, alter it in some fashion (obeying the laws of algebra), so that it no longer has the $\infty-\infty$ form.
So for the opening example in your post, the mathematical strategy, as you say, is to rationalize the radicals. And you have probably learned other $\infty-\infty$ examples with different mathematical strategies.
