Consider the following function of t, f(t)=1/a[u(t-T0)-u(t-T0-a)]. Here both T0 and a are positive numbers. If you pick some value for T0 and a you discover that you have a rectangular pulse starting at T0 having width a and height 1/a. Obtain the Laplace transform of this pulse by direct integration. If you let a -> 0 you obtain a unit impulse. Let a -> 0 in the expression you obtained for the laplace transform of the pulse to obtain laplace transform of the shifted unit impulse.