Problem

Let

and

What is

Solution

We are not given any information about f beyond that fact that it has a limit as x approaches 3. If we are to compute the composition g(f(x)), f must be in the domain of g in some interval containing 3, even though it need not be defined at x = 3. Since all real numbers are in the domain of g, anything that works to establish the limit of f as x approaches 3 is in the domain of g.

As x approaches 3, f(x) approaches 5, so we can write

Since g(x) is a polynomial, the limit on the right hand side can be evaluated by substituting 5 for u.