This is from a an AP Calculus BC 2006 exam.
The answer is
My question is, why start at t = 2? Why add the initial value?
I wrote my improper integral from 0 to infinity.
Well, you are only given the derivatives of $\displaystyle x(t)$ and $\displaystyle y(t)$, 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 $\displaystyle y(0)=k$, but this case you are given the initial condition is $\displaystyle y(2)=-3$. So to find the horizontal asymptote, you have to find the limit of the function (which will be the integral of the derivative) from $\displaystyle t=2$, because we know at $\displaystyle y(2)=-3$. 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.