Polynomial and Rational Functions
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
There are certain behaviors of rational functions that give us clues about their limits. For the limits of rational functions, we look at the degrees of their quotient functions, whether the degree of the numerator function is less than, equal to, or greater than the degree of the denominator function. These characteristics will determine the behavior of the limits of rational functions.
I want talk about limits of rational functions and by that I mean what happens as x gets larger and larger?
First let's look at two really simple examples; f of x equals 10 over x. What happens as x gets larger and larger? Let's start with a number like 1, 10 over 1 would be 10, how about 10? 10 over 10 is 1. 100, 10 over 100 0.1, how about a 1000? 10 over a 1000 is 0.01. You can see that as x gets larger, that the values of f of x are going to zero. We say as x goes to infinity f of x goes to 0 or we write limit as x approaches infinity of f of x equals 0. This is a kind of notation that you'll see a lot in Calculus.
Let's see what happens to this example, g of x equals 25 over x squared. We'll start again with x=1 we get 25 over 1 squared which is 25, 10 we get 25 over 100 which is 0.25, 100 a 100 squared is 10,000 25 over 10,000 is 0.0025 and you can see that as x is going to infinity, this is going to 0 even faster than the other function and so here again we say limit as x approaches infinity of g of x is 0. Now you can probably gather that this sort of thing is always going to happen. When you have a function like this h of x equals a constant over x to a positive integer power, if that's the kind of a function you have, then the limit as x approaches infinity of h of x is always 0 and it turns out that the limit as x approaches negative infinity of h of x is always 0 and so those are really important facts and we're going to use those to find limits of other rational functions.