The intuitive definition of limit is that for any y-range around a limit point there is an x-range so the graph exits through the sides of the box rather than though the top or bottom. One practical way to get an idea if a point is a limit is to find values of delta that work when epsilon in .1, .01, and .001.
The Function choice slider allows you to work through a series of functions where something interesting happens at x=0, y=1.