A limit is a point on a graph that we look at from the left and the right and we see the numbers approaching the point on the graph we looked at. We can tell if a limit exists by looking at the point and seeing if the left hand numbers and the right hand numbers approach the same number. Limits help us explain function behavior at points of discontinuity by showing what happens at the right side and the left side at the point of discontinuity.
The connection between factors and zeros is that when its all done the factors and the zeros are the same number with just opposite. For example if one factor is positive the zero will be negative. Division helps us factor the polynomials so we can get the factors. How ever many degrees the polynomial has thats how many zeros it will have. And that does tell us the number of factors because how ever many degrees it has is how many factors it will have.
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