59 Let (x) = x sin(1/x). x#0 0. X=0 and g(x) = x/(x). The graphs of / and g are indicated in the figures below. = X =x2 DIFFERENTIATION (a) Show that f and g are both continuous at 0. (b) Show that f is not differentiable at 0. (c) Show that g is differentiable at 0 and give g'(0). (Important). By definition f (eth) - f(c) f(c) = lim h provided this limit exists. Setting x = c + h, we can write (3.1.5) f(c) = lim f(x) - f(c) I+C x - C This is an alternative definition of derivative which has advan- tages in certain situations. Convince yourself of the equivalence