Question 2 (4 points) 1. (3 points) Let f be defined on [0, co) and integrable on all [0, t] where t > 0. Also suppose that lim f(x) = L E R. Show that X-+00 lim x-+00 X Jo f (t) dt = L (Hint: Here is the intuitive idea: for large x (say, x 2 X) we have f (x) ~ L, then I f (t)dt ~ L(x - xo), and - Sof(t) dt = =ff(t )dt + L(x -xo) - 0+ L. Now make this precise.) (Note the quantity - Jf (t) dt is called the average of f on [0, x].) 2. (1 point) In the situation of 1. conclude that if L # 0 then J f (x) dx diverges. (Note: the converse is false, the integral may diverge even if L = 0, see f (x) = = 1 also 1+x the limit may not exist and the integral could still converge. This only says: if the limit exists and is finite, and if the integral converges, then the limit must be zero. There are also quicker arguments than using 1.)