Say we have this rule of limit: $$\lim_{x\to a} (f(x) + g(x)) = \lim_{x\to a}f(x) + \lim_{x\to a} g(x)$$
It is implied that we can do this only if both limits exist individually.
My questions is why we can't do this if only 1 limit is nonexistent(or infinity) and the other one is a finite limit.
It still doesn't form an indeterminable form so I am confused as to why we can't do it.
You can write it notationally, but the limits on both sides will not exist because $\lim f(x)+g(x)$ exists only if both of $\lim f$ and $\lim g$ exist, or neither of them.
Suppose $\lim_{x\rightarrow a} f(x)$ and $\lim_{x\rightarrow a}f(x)+g(x)$ exist. Then $\lim_{x\rightarrow a}([f(x)+g(x)]-f(x))$ exists and equals $\lim_{x\rightarrow a}[f(x)+g(x)]-\lim_{x\rightarrow a} f(x)$ by the same theorem you quoted. That is, $$\lim_{x\rightarrow a}([f(x)+g(x)]-f(x))=\lim_{x\rightarrow a}[f(x)+g(x)]-\lim_{x\rightarrow a} f(x)$ $$ $$\lim_{x\rightarrow a}(g(x))=\lim_{x\rightarrow a}[f(x)+g(x)]-\lim_{x\rightarrow a} f(x)$ $$
We have proven that the limit of $g(x)$ exists, so by the contraposative, we cannot separate them if one exists and one does not.
