# A limit problem

by suffian
Tags: limit, solved
 P: n/a I just need some help showing how this limit systematically follows from the limit rules:  # x 2 1 # t 1 lim --- # ---------- dt = --- x->0 3 # 2 3 x # 0 t + 1 My first chain of thought led to breaking the expression up as follows: 1/x^2 * ( 1/x * Integral[0..x, t^2/(t^2+1)] ) Then I just kind of figured that the subexpression on the right was the average value of the function being integrated from 0..x and as x->0 the average value would approach x^2/(x^2+1), which led to: 1/x^2 * x^2/(x^2+1) = 1/(x^2+1) which would approach one as x approached zero. But clearly that's wrong (not surprisingly since I made a sketchy move in the middle) since the answer is one-third. Can anyone show me how to do this? edit: possibly w/o actually integrating because this is an exercise in which you're expected to know the FTofC but not how to integrate that. edit2: oh, not supposed to no l'hospital's rule either.