General question: Am I allowed to take the limit of a single variable in a multivariable function with a domain restrictions to obtain a new function instead of a value?
Example: $$f(x,y,a_1,a_2) = a_1x^2+a_2y^2$$ $$\lim_{a1 \to 0^+}f(x,y,a_1,a_2)$$ Where $a_1$ and $a_2$ must be positive and NON zero (This type of domain restriction is necessary to the type of problem I am working on)
Generally this would be just a function of $x$ and $y$ with constant coefficients $a_1$ and $a_2$. But I'd like to explore different values of $a_1$ and $a_2$, thus the function being $f(x,y,a_1,a_2)$
With that being said, would the limit of the above function be the following and would I be able to go ahead and use this new function in further operations and analysis?$$\lim_{a1 \to 0^+}f(x,y,a_1,a_2) = a_2y^2$$
My reasoning for this to be true is the fact that a right-hand limit is not inclusive of the value and just $0+\epsilon$ and therefore not in violation of my domain restrictions of $a_1$
Additional question: Would I also be able to conclude that the new function does not depend on $x$ at the zero limit of $a_1$?
I don't see anything wrong with what you have said. However, in general one has to be quite careful when dealing with taking limits of functions, since there are many different types of convergence (pointwise, uniform, norm etc.), and taking limits may not interact nicely with other things like integration (see Lebesgue dominated convergence).
As you have noticed, $\lim_{a_1 \to 0^+} a_1 x^2 = 0\,$, (if we understand this as a pointwise limit of functions, since this is not true uniformly or in norm on $\mathbf{R}$) and this shows that your set of parameters is not closed (taking a limit resulted in you leaving the parameter space), and your new function is independent of $x$. If you really insist that $a_1>0$, then maybe taking this limit is not a good idea!