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? 

With the Fundamental theorem of calculus, you can express
\begin{align}
&\int_0^a f(x) \, dx = F(a)-F(0)
\\ \Leftrightarrow &\frac1a\int_0^a f(x) \, dx = \frac{F(a)-F(0)}{a}=G(a)
\end{align}
If you now want to evaluate $\lim_{a\rightarrow 0} G(a)$ you get
\begin{align}
\lim_{a\rightarrow 0} G(a) =\lim_{a\rightarrow 0} \frac{F(a)-F(0)}{a}= ?
\end{align}
Does the above formulation remind you of something?
