Improper integrals with parameters

[MZD859]

Homework Statement

Let f be continuous on the interval [0,inf] with abs(f(x)) < M, set
F(u) = \int \frac{u*f(x)}{u^2 + x^2} integral from 0 to inf (bad with latex)

prove that lim F(u) as u --> 0 = f(0)

Homework Equations

none

The Attempt at a Solution

I split the integral into the intervals from (0,1) and (1,inf) since (1,inf) is controlled

now I can't seem to work out the first term. Any help would be great as I'm totally stuck

 

[lanedance]

qualitatively, how about splitting it into (0,e) and (e,inf), then show for any e>0, you can choose u close enough to zero that the 2nd term goes to zero, then consider the limit as e->0 using the continuity of f