Assume a function $f(x,y)$ over the real numbers that is computable in the sense of computable analysis. Let $g(x) = \lim_{y \to \infty}f(x,y)$. I have two questions:

1) Under what conditions does $g(x)$ exist?

2) Assume $g(x)$ exist? Is it always computable? If not, then under what conditions it is computable?