I have solved Question 3 on the MATH 100 V01 Midterm exam.

a) I have memorized the definition of continuity. When the limit of g(x) as x approaches a should reach a number of g(a), because it is getting extremely close to a, so that we say that g(x) is equal to g(a) when x goes to a.

b) Same concept is applied in this question as part a. Using the definition of continuity, in order to make f(w) to be continuous at g(a), the limit of f(w) as w goes to g(a) should go to f(g(a)).

c) Recall part (a) and (b). We can say that the limit of g(x) as x approaches a is also equal to f(g(a)) due to the inter relationship between the functions. Therefore, we can say that the limit of f(w) as w goes to a is equal to f(g(a)) which is equal to the limit of f(g(x)) as x approaches a.