Well, you are only given the derivatives of and , so there are an infinite number of possible functions with these derivatives. So you need an initial value condition to determine which function you are dealing with. Usually a problem gives you the initial function of , but this case you are given the initial condition is . So to find the horizontal asymptote, you have to find the limit of the function (which will be the integral of the derivative) from , because we know at . If you weren't given an initial condition, there'd be an infinite number of functions with these derivatives, and therefore there'd be an infinite amount of respective horizontal asymptotes.