Limits of Composite Function

No, you don't need the extra limit on the outside. 

lim x->c g(x) is going to be some constant number, provided that the limit exists. For the sake of my argument, let's suppose that limit is 4.

So, lim x->c f( lim x->c g(x) ) is the same thing as lim x->c f(4), which is just f(4). After finding the inside limit, x is no longer part of the expression, so the outside limit isn't necessary.

Hope that makes sense!