Perhaps you're just kidding, or I misunderstand something. Isn't the Dirac Delta *defined* such that the int_-inf^+inf(delta(t-T)dt = 1.0??? And since the Dirac delta functions value is *defined* as zero everywhere except when t-T = 0, then the lim M->0 int_(T-M)^(T+M) delta(t-T)dt is also equal to 1.0 ???
The partial integral from -inf to (T-M) as lim M->0 remains 0.0 as well as the partial integral from (T+M) to +inf remains 0.0. So the difference between lim M->0 int_(T-M)^(T+M) delta(t-T)dt and int_-inf^+inf delta(t-T)dt is zero (because M never reaches exactly 0).