Another important class of limit problems we will need to deal with over the next few weeks have to do with the behavior of functions as their independent variables become either large positive or large negative. Many such problems involve the behavior of polynomials, so it is worth our while to spend some time thinking about polynomials at infinity.
Here is a typical example.
A plot shows the behavior of this function near the origin:
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Note the difference in the horizontal and the vertical scales used for this plot. Had we done this plot with equal scales the behavior would have been even more dramatic. As x moves away from the origin in either direction the function becomes large negative and positive. We indicate this by writing
and
Is there an easy way to see why this should be the case? Yes, there is. All we have to do to predict the behavior of a polynomial at infinity is to look at the behavior of its first term. As x gets very large positive or negative, the first term in the sum becomes so much larger than the other terms that it gets to determine the overall behavior of the polynomial.
There are a couple of ways to demonstrate this. The first is to plot only the first term in the polynomial alongside the entire polynomial. This is what you get when you do this for the present example:
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The curve that does not pass through the origin is the original polynomial. The picture shows that although the two differ near the origin, as we move further away they look more and more alike. This effect is even more dramatic when we zoom out.
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Another way to see that the first term dominates is to rewrite the polynomial with the first term factored out.
What happens to the terms inside the parentheses as x gets large positive or negative? With the exception of the 1, all the other terms go to 0 in the limit of large x. This is yet another way of seeing that the first term in a polynomial (i. e. the term with the largest power of x) dominates the behavior of the polynomial at 
.
The last thing we need to pay attention to when determining the behavior at 
is whether the power of x is even or odd. Even powers produce large positive results for both large positive and large negative x, while odd powers give large positive when x is large positive, and large negative when x is large negative. Another simple plot will demonstrate this clearly.
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The x2 gets large positive in both directions, while the other polynomial gets large negative in the negative direction and large positive in the positive direction.
What does this one do?
The dominate term has an even exponent, so we would expect this to get large positive in both directions. However, the presence of the negative coefficient on the leading term turns this around. In both directions this polynomial will become large negative.
Before we leave the subject of polynomials at infinity, there is one last thing we should make note of. Examine the last plot again. Something about that plot which is very striking is the relative rates with which the two polynomials get large as x goes to 
. The x5 term grows much more quickly than the x2 term. Generally speaking, terms with larger exponents will grow more quickly than terms with smaller exponents. This effect becomes important in the next section.
A class of functions that we will encounter off and on over the next few weeks is the class of rational functions. A rational function is a ratio of two polynomial functions. A typical example is
Here is a plot of this function.
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In the study of limits, rational functions are interesting for two reasons. They tend to be undefined at a number of places (anywhere that the polynomial in the denominator is 0) and they have an interesting range of behaviors as x gets large in either direction.
The denominator can sometimes vanish, which makes for interesting behavior. The denominator in the example above vanishes both at 1 and -1, although a cancelation effect makes the function not blow up at x = -1. The behavior near 1 is more typical. As x approaches 1, the denominator becomes very small, while the numerator has a value of about 2. Consequently, the ratio becomes larger and larger as x approaches 1.
The general principle here is that rational functions tend to ‘blow up’ as x approaches points where the denominator is 0. The only detail that varies from example to example is exactly what direction the function blows up in. We can of course see what happens by plotting the function, but we would like to be able to figure this out with the aid of advanced technology. The easiest way to predict what direction a rational function will blow up in is to reason something like the following. In the example above, it is clear that the denominator will vanish at x = 1. What happens when we plug in an x that is just slightly smaller than 1? Roughly speaking, we get this:
When we raise a number slightly smaller than 1 to a positive power we get a number which is even smaller. Thus our number will be
The denominator will end up being a small negative number, while the numerator will be slightly smaller than 2. The closer we get to 1, the smaller the denominator gets, and hence the larger negative the whole thing becomes. We summarize by saying
For numbers to right of 1, the reasoning goes like this:
To summarize, when a function blows up as it approaches some finite point we say that it approaches a vertical asymptote. To give a complete description of a function in the neighborhood of a vertical asymptote we have to compute limits on either side of the point in interest. In this case we learned that
Some rational functions have finite limits as x gets very large positive or negative. Here is an example.
The easiest way to see what this function will do at infinity is to divide the numerator and the denominator by the leading term of the numerator.
Similar reasoning holds for the negative direction.
When a function has a finite limit as it approaches either positive or negative infinity, we say that the function approaches a horizontal asymptote.
Section 2.6: 4, 6, 7, 19, 20, 28, 29, 39, 40