Exercise II.3.5 (c) in Hartshorne, Algebraic Geometry, asks to find an example of a surjective, finite-type and quasi-finite morphism of schemes which is not finite.

I need to find a finitely generated $A$-algebra $B$ which is not finite generated as an $A$-module. The only examples, I could find, of such a kind of $B$ give rise to a morphism which is not quasi-finite. Basically I was trying to use some modification of the classic $B=\mathbb{C}[x]$. I have also thought to find a morphism which is not closed, since we know that a finite morphism is always closed, but even this way didn't lead me anywhere.

Do you have any suggestion?

P.S.: $f$ quasi-finite means that $f^{-1}(y)$ is a finite set for every point $y\in Y$.

MOREOVER: while thinking at this example, I asked another question to myself. Which is a quasi-finite morphism which is not of finite-type? 

Thank you very much!