Let $f$ be a continuous function on [$0, 1$] with $f(0) =1$. Let $ G(a) = 1/a ∫_0^a f(x)\,dx$ then which of the followings are true? Let $f$ be a continuous function on [$0, 1$] with $f(0) =1$. Let $ G(a) = 1/a ∫_0^af(x)\,dx$ then which of the followings are true?  


*

*$\lim_{(a\to 0)} G(a)=1/2$   

*$\lim_{(a\to0)} G(a)=1$  

*$\lim_{(a\to 0)} G(a)=0$  

*The limit $\lim_{(a\to 0)G(a)}$ does not exist.   


I am completely stuck on it. How should I solve this? 

Hint: Use l'Hospital's rule for $\lim_{a\to 0}G(a)$. We see that $G(a)\to 1$ when $a$ tends to zero.
