1. The problem statement, all variables and given/known data Sketch sin (1/x), x sin (1/x) and x^2 sin (1/x) and show they are discontinuous at x=0. 3. The attempt at a solution I *know* how to do this problem with the usual curve sketching methods (important points, asymptotes, turning points, "tends to") but my book has a method that involves breaking the compound function into smaller ones. I don't know how to break x sin (1/x) into compound functions, though. I don't think it's necessary to totally break it down anyway since the graphs all have about the same shape. I can do sin (1/x)... I think... do check my solution too. y = f(x) = sin (1/x) w = g(x) = 1/x u = h(x) = sin w sin w can reach 1 when w = π/2, 5π/2... and -1 when w = 3π/2, 7π/2... and is 0 for any integral multiple of π. Substituting for x, sin (1/x) reaches 1 when x = 2/π, 2/5π... -1 when x = 2/3π, 2/7π... 0 for x = 1/kπ and k is any integer. Deduce that f(x) tends to slightly below 0 at negative infinity and slightly above 0 at positive infinity. Also substitute values for x = 1/kπ: 1/π, 1/2π, 1/3π, 1/4π When plotting a graph the intervals between successive 0s become smaller to negligible. Formula for 1, -1 also have similar form, so values for -1, 1 and 0 cluster as x tends towards 0 (or k tends towards infinity). By inspection the values oscillate between 1, 0 and -1, and that describes the graph of sin (1/x). Graph is discontinuous for 1/x at x=0. I could do the rest using calculus, but I'm told there may be no need to (or told this so that I have to do everything the long way). The values for 0 should be at the same value for x in all 3 equations. I *know* that the y values of the turning points are not the same, but that's after differentiation.