3. When examining a function to determine if it has a limit, it is sometimes convenient to make a change of variable. The theorem you will be asked to prove is designed to assist in using the change of variable in limits of functions. (a). Prove the following theorem: Theorem 0.1 Suppose f AR, g B A. Suppose that a is an accumulation point of A, b is an accumulation point of B, I. lim bg(t) = a, II. there is a neighborhood Q of b such that for t = QNB, g(t) a, and III. f has a limit at a. Then fog has a limit at b and lim f(t) = lim f(g(t)). x-a tb (1) (b). Consider f: R \ {0} R defined by by sin(x) f(x) = " x 0. (2) A standard argument in calculus shows that lim sin(x) = 1. x 0 X (3) You may use this limit. Use the theorem you just proved to show that the function h(x) = cos(x) = x T " x- has a limit at /2, and find that limit. (4)