Question 4 Let Q be defined by Q(x) = ( x 2 sin(1/x) if x = 0 0 if x = 0 Find the error in the following argument showing that Q (0) = 0. Your answer should explain the problem and indicate how to fix it. (The statement is true, but the proof given is wrong.)

Proof: According to Definition 3.1.1 it must be shown 0 = lim h0 Q(h) Q(0) h = lim h0 Q(h) h = lim h0 h 2 sin(1/h) h = lim h0 h sin(1/h). Now use Theorem 2.5.1 with g(x) = x, f(x) = x sin(1/x) and h(x) = x. Note that since 1 sin(1/x) 1 (1) if x = 0 it follows that, multiplying each term of Inequality (1) by x, that h(x) = x x sin(1/x) = f(x) x = g(x) (2) for all x = 0. Since limx0 h(x) = 0 = limx0 g(x) it follows from Theorem 2.5.1 that lim h0 h sin(1/h) = lim h0 f(h) = 0