I have just learned that if $\lim \limits_{(x,y) \to (a,b)}f(x,y)=f(a,b)$ then the function is continuous at that point.
My question is, let's say we have a very simple function and limit, such as:
$\lim \limits_{(x,y) \to (2,3)}x^2+y^2=L$
Can I, without any further justification, just "plug in" (2,3) to get L=13? Or would I need some further proof (like the squeeze theorem). Similarly, if I can just "plug in" x and y values into a function and get a real number back, is that sufficient to conclude that the function is continuous at that point?
Also please avoid using epsilon delta stuff, I haven't learned that at all.
No. By "plugging in numbers," what you are assuming is continuity at the point you're plugging in. If you have already shown your function is continuous (or you are allowed to assume this), then you're fine, otherwise, you need to actually show that the limit equals the value you get by plugging in those numbers. The way one traditionally does this is by "epsilon-delta stuff."