If your function has an anti-derivative that is continuous, then you can use the property that lim f(x) for x approaching a = f(a). If your integral is f(x)dx and it has an anti-derivative F(x), then you can use this known property of limits as applied to continuous functions.

Basically continuity is defined in analysis by imposing the condition that if you have a function in some open interval (a,b) and the limit of x->c f(x) = f(c) for all values in that interval, then the function is continuous (but the limit from both sides has to give the same value and it has to be finite).

Now for a limit approaching infinity or negative infinity we only consider a one-sided limit, but this should exist if your anti-derivative is continuous.

That's the basic intuition of why you can use lim F(a) as a -> infinity if f(x) is a Riemann-Integrable function over some interval.