Yes, there should be a theorem in your book that says that if and , then .
There should also be theorems that say that, under the same hypotheses,
and, provided g(x) is not 0 in some neighborhood of a,
There is another limit theorem that is not emphasised nearly enough:
If f(x)= g(x) for all x except a, the
So that if, for example for all x and for all x except x= 1 and g(1)= 1000, then .