Suppose we want to prove $\lim\limits_{t\rightarrow\infty} x(t)=0$ where $x(t)=e^{-at}\int_{t_0}^{t}e^{as}r(s)ds+C$, $a>0$ and $\lim\limits_{t\rightarrow\infty} r(t)=0$.
I first did $\lim\limits_{t\rightarrow\infty} x(t)=\dfrac{\lim\limits_{t\rightarrow\infty} \int_{t_0}^{t}e^{as}r(s)ds+C}{\lim\limits_{t\rightarrow\infty}e^{at}}=\dfrac{\lim\limits_{t\rightarrow\infty} \int_{t_0}^{t}e^{as}r(s)ds}{\lim\limits_{t\rightarrow\infty}e^{at}}$ and tried to use L'Hôpital but am having trouble. I know that the denominator goes to $\infty$ but how to do the top? If I can get the top to go to $\infty$ then applying L'Hôpital would give me the desired result. Any help please?