In this explainer, we will learn how to evaluate limits of a function when tends to infinity and explore unbounded limits that tend to infinity as approaches a certain value.
By definition, we will say that , where is a real number provided that the values get as close as we like to and that is big enough.
Likewise, means that for all that are large enough but negative.
Before looking at examples of convergence, let us note that there are essentially two things that can go wrong with convergence as goes to (or ):
- The function oscillates too much, so there is no number . This is the case with .
- The values increase (or decrease) without bound as the positive or negative value of gets larger and larger.
In the figure shown, we have the following:
(a) is the graph of which goes to as the positive value of gets larger, while it goes to as the negative value of gets larger. We write these as
(b) is the graph of the polynomial function from which we see that, like , the values become infinite in absolute value as also becomes infinite, but in the opposite sense to :
(c) is the graph of the exponential function , where the graph suggests strongly that
In fact, in order to determine the behavior of limits at infinity of the two classes polynomials and rational functions, it is enough to know the following:
- Infinite limits behave with respect to sums and products as expected, so that, for example, if , and , then Differences and quotients are not as straightforward (see below). Corresponding facts hold for .
- The following is how infinite limits interact with constants. If , then For any real number , Corresponding facts hold for .
- Limits that would evaluate to or or (whether or ) are all indeterminate and require some algebraic simplification if they can be evaluated at all.
As a result, we have because this is the product of and . In fact, since , we see from property (3) that and then by property (4). We also have that because which has limit as .
In order to determine the limit , we proceed, as with all rational functions, by dividing both the numerator and the denominator by the highest power of visible in both these polynomials: so that
Example 1: Finding the Limit of a Polynomial Function at Infinity
We factor out , the highest power of :
Notice that we choose to factor the polynomial rather that argue by summands. We could have started by saying that , , and , but then what is
And the answer is that we cannot decide, because is an indeterminate form.
Of course, the algebra merely affirms the intuition that the leading term determines where the polynomial goes as tends to infinity. The term dominates when is large.
Example 2: Finding the Limit of a Rational Function at Infinity
Following the strategy for polynomials, factor out leading terms first, and then cancel those:
Now we can “substitute” for the limits:
Next is an alternative strategy.
Example 3: Finding the Limit of a Rational Function at Infinity
First, note that the numerator is a polynomial of degree 4 and the denominator is a polynomial of degree 3.
As the degree of the numerator is greater than the degree of the denominator, the limit will be infinite.
Since the leading coefficient in the numerator is negative (the coefficient of ), therefore the limit is .
And the following is another way.
Example 4: Finding the Limit of a Rational Function at Infinity
If , and is a constant, then . Also, if is a constant, then . Use these facts, along with the fact that dividing both the numerator and the denominator of a fraction by the same quantity results in an equivalent fraction, to find as follows: