Having trouble knowing when to apply dominated convergence

Hey yall, I want to use dominated convergence on an integral, namely lim_eps->0 int_R k_eps(x) dx, where k_eps:=|1/eps|/(pi•((x/eps)2 +1)), and epsilon is a positive real. I know that because of the potential singularity in the integrand at 0, ill have trouble bounding the function. After preforming a u-sub:u=(x/eps) we arrive at int_R 1/(pi•((u)2 +1) du, where there are jo dependencies on eps. Now I can show the integrand is bounded in between the range values of 0, and 1/pi. At this point, since I can show it is bounded, that satisfies the criteria for DCT. Im concerned that my manipulation via u-sub cant be used to show criteria matching for DCT though. A professor told me that a u-sub only changes how the integral looks, not the integral itself, so in that philosophy, this is a valid manipulation. Thanks for all the help!

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