On-line Math 21

1.3.1  Examples of computing limits using rules and theorems.

Problems with hints

Problem 1 Compute

lim x® 3  x2-9 x2-2x-3 .

Did I factor it correctly?

You have to remember the standard tricks for factoring polynomials. Always start out hoping it will factor easily, which in this case means that you can factor things with integers. If so, then the integers you use have to be the factors of the constant term. The denominator, x²-2x-3 , can be factored by thinking about the factors of -3 (which are just 3 and -1 ). So, the factors of x²-2x-3 can only be (x+1) and (x-3) , because you take (x-factor) when trying out this method. Well, it works in this case. In general you have to try several ways of dividing up the constant term, but it's pretty easy once you get used to it.

The numerator is more standard. It is a difference of squares. Always look for that, since you can easily factor a difference of squares, x²-a² = (x+a)(x-a) . Here, a = 3 .

Then put the factored numerator, and denominator, together.

x2-9 x2-2x-3 = (x+3)(x-3) (x-3)(x+1) .

What next?

What's the answer?

By David L. Johnson, last modified 3/10/00.