Question

Solution to (a). Let f(x) =(0, x 6= 0 1 x = 0 and g(x) =(0, x 6= 0 2 x = 0 . Then limx→0 f(x) = 0 and limx→0 g(x) = 0. However, if x 6= 0, then g(f(x)) = g(0) = 2, so that limx→0 g(f(x)) = 2 6= 0. Solution to (b). If both f and g are continuous, then f(p) = q and g(q) = r. The limit limx→p g(f(x)) = r = g(f(p)) holds by Theorem 4.3.9. Solution to (c). Assume f is continuous. Then q = f(p). Let ε > 0. there is a ρ > 0 such that |g(x) − r| < ε whenever 0 < |x − q| < ρ. By the continuity of f, there is a δ > 0 such that |f(x)−f(p)| < ρ whenever |x−p| < δ. Hence, if 0 < |x−p| < δ, then |f(x)−f(p)| = |f(x)−q| < ε, so that |g(f(x))−r| < ε. This shows that limx→p g(f(x)) = r. If g is continuous, then r = g(q). We have then limx→p g(f(x)) = g(limx→p f(x)) = g(q) = r.