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? 

Note that $G(a)$ is the mean (or average) value of the function on the interval $[0,a]$. Here’s an intuitive argument that should help you see what’s going on. The function $f$ is continuous, and $f(0)=1$, so when $x$ is very close to $0$, $f(x)$ must be close to $1$. Thus, for $a$ close to $0$, $f(x)$ should be close to $1$ for every $x\in[0,a]$, and therefore its mean value should also be close to $1$.
From that it should be easy to pick out the right answer, but it would also be a good exercise for you to try to prove that the answer really is right.
