Dimension of integral schemes of locally finite type over a field

Question

In Exercise 3.20 of Algebraic Geometry, Hartshorne makes several claims about the dimension of an integral scheme of finite type over a field. For instance, he claims that the dimension is equal to the dimension of the local ring at any closed point. I believe I understand how to solve this exercise, but I don't see why why need the scheme to be finite type, as opposed to merely locally finite type. Could anyone give me an example of such a scheme of locally-finite type, but not of finite type, such that its dimension is not equal to the dimension of its localization at a closed point?