Explainer: Limits at Infinity and Unbounded Limits

In this explainer, we will learn how to evaluate limits of a function when ๐‘ฅ tends to infinity and explore unbounded limits that tend to infinity as ๐‘ฅ approaches a certain value.

By definition, we will say that lim๏—โ†’โˆž๐‘“(๐‘ฅ)=๐ฟ, where ๐ฟ is a real number provided that the values ๐‘“(๐‘ฅ) get as close as we like to ๐ฟ and that ๐‘ฅ is big enough.

Likewise, lim๏—โ†’๏Šฑโˆž๐‘“(๐‘ฅ)=๐ฟ means that ๐‘“(๐‘ฅ)โ‰ˆ๐ฟ for all ๐‘ฅ 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 ๐‘ฅ goes to โˆž (or โˆ’โˆž):

  • The function oscillates too much, so there is no number ๐ฟ. This is the case with ๐‘“(๐‘ฅ)=๐‘ฅsin.
  • The values ๐‘“(๐‘ฅ) increase (or decrease) without bound as the positive or negative value of ๐‘ฅ gets larger and larger.

In the figure shown, we have the following:

(a) is the graph of ๐‘“(๐‘ฅ)=๐‘ฅ which goes to +โˆž as the positive value of ๐‘ฅ gets larger, while it goes to โˆ’โˆž as the negative value of ๐‘ฅ gets larger. We write these as limandlim๏—โ†’โˆž๏—โ†’๏Šฑโˆž๐‘ฅ=+โˆž๐‘ฅ=โˆ’โˆž.

(b) is the graph of the polynomial function ๐‘(๐‘ฅ)=โˆ’143,500(๐‘ฅ+10)(๐‘ฅ+1)(๐‘ฅโˆ’20)(๐‘ฅโˆ’22)(๐‘ฅโˆ’30) from which we see that, like ๐‘“(๐‘ฅ)=๐‘ฅ, the values become infinite in absolute value as |๐‘ฅ| also becomes infinite, but in the opposite sense to ๐‘“: limandlim๏—โ†’โˆž๏—โ†’๏Šฑโˆž๐‘(๐‘ฅ)=โˆ’โˆž๐‘(๐‘ฅ)=+โˆž.

(c) is the graph of the exponential function ๐‘”(๐‘ฅ)=2๏—, where the graph suggests strongly that limandlim๏—โ†’โˆž๏—๏—โ†’๏Šฑโˆž๏—2=+โˆž2=0.

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:

  1. lim๏—โ†’โˆž๐‘ฅ=+โˆž.
  2. lim๏—โ†’๏Šฑโˆž๐‘ฅ=โˆ’โˆž.
  3. Infinite limits behave with respect to sums and products as expected, so that, for example, if lim๏—โ†’โˆž๐‘“(๐‘ฅ)=+โˆž, and lim๏—โ†’โˆž๐‘”(๐‘ฅ)=+โˆž, then limlim๏—โ†’โˆž๏—โ†’โˆž๐‘“(๐‘ฅ)+๐‘”(๐‘ฅ)=+โˆž,๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)=+โˆž. Differences and quotients are not as straightforward (see below). Corresponding facts hold for โˆ’โˆž.
  4. The following is how infinite limits interact with constants. If lim๏—โ†’โˆž๐‘“(๐‘ฅ)=+โˆž, then limiflimif๏—โ†’โˆž๏—โ†’โˆž๐ด๐‘“(๐‘ฅ)=+โˆž๐ด>0,๐ด๐‘“(๐‘ฅ)=โˆ’โˆž๐ด<0. For any real number ๐ด, limlim๏—โ†’โˆž๏—โ†’โˆž(๐‘“(๐‘ฅ)+๐ด)=+โˆž๐ด๐‘“(๐‘ฅ)=0. Corresponding facts hold for โˆ’โˆž.
  5. Limits that would evaluate to โˆžโˆ’โˆž or 0โ‹…โˆž or โˆžโˆž (whether +โˆž or โˆ’โˆž) are all indeterminate and require some algebraic simplification if they can be evaluated at all.

As a result, we have lim๏—โ†’โˆžโˆ’3๐‘ฅ=0 because this is the product of (โˆ’3) and 0=1๐‘ฅlim๏—โ†’โˆž. In fact, since lim๏—โ†’๏Šฑโˆž๐‘ฅ=โˆ’โˆž, we see from property (3) that lim๏—โ†’๏Šฑโˆž๏Šฉ๐‘ฅ=โˆ’โˆž, and then lim๏—โ†’๏Šฑโˆž๏Šฉโˆ’3๐‘ฅ=0 by property (4). We also have that lim๏—โ†’๏Šฑโˆž๏Šฉ2๐‘ฅโˆ’3๐‘ฅโˆ’70=โˆ’โˆž, because 2๐‘ฅโˆ’3๐‘ฅโˆ’70=๐‘ฅ๏€ผ2โˆ’3๐‘ฅโˆ’70๐‘ฅ๏ˆ๏Šฉ๏Šฉ๏Šจ๏Šฉ which has limit (โˆ’โˆž)ร—(2)=โˆ’โˆž as ๐‘ฅโ†’โˆ’โˆž.

In order to determine the limit lim๏—โ†’โˆž2โˆ’๐‘ฅ3๐‘ฅ+1, we proceed, as with all rational functions, by dividing both the numerator and the denominator by the highest power of ๐‘ฅ visible in both these polynomials: 2โˆ’๐‘ฅ3๐‘ฅ+1=โˆ’+=โˆ’13+๏Šจ๏—๏—๏—๏Šฉ๏—๏—๏Šง๏—๏Šจ๏—๏Šง๏— so that limlimlimlim๏—โ†’โˆž๏—โ†’โˆž๏Šจ๏—๏Šง๏—๏—โ†’โˆž๏—โ†’โˆž2โˆ’๐‘ฅ3๐‘ฅ+1=โˆ’13+=2๐‘ฅโˆ’13+1๐‘ฅ=0โˆ’13+0=โˆ’13.

Example 1: Finding the Limit of a Polynomial Function at Infinity

Find lim๏—โ†’โˆž๏Šฉ๏Šจ๏€น2๐‘ฅโˆ’4๐‘ฅโˆ’3๐‘ฅโˆ’7๏….

Answer

We factor out ๐‘ฅ๏Šฉ, the highest power of ๐‘ฅ: 2๐‘ฅโˆ’4๐‘ฅโˆ’3๐‘ฅโˆ’7=๏€น๐‘ฅ๏…๏€ผ2โˆ’4๐‘ฅโˆ’3๐‘ฅโˆ’7๐‘ฅ๏ˆ.๏Šฉ๏Šจ๏Šฉ๏Šจ๏Šฉ

Now lim๏—โ†’โˆž๏Šฉ๐‘ฅ=+โˆž while lim๏—โ†’โˆž๏Šจ๏Šฉ๏€ผ2โˆ’4๐‘ฅโˆ’3๐‘ฅโˆ’7๐‘ฅ๏ˆ=2โˆ’0โˆ’0โˆ’0=2.

Therefore, limlimlim๏—โ†’โˆž๏Šฉ๏Šจ๏—โ†’โˆž๏Šฉ๏—โ†’โˆž๏Šจ๏Šฉ2๐‘ฅโˆ’4๐‘ฅโˆ’3๐‘ฅโˆ’7=๏€น๐‘ฅ๏…๏€ผ2โˆ’4๐‘ฅโˆ’3๐‘ฅโˆ’7๐‘ฅ๏ˆ=(+โˆž)(2)=+โˆž.

Notice that we choose to factor the polynomial rather that argue by summands. We could have started by saying that lim๏—โ†’โˆž๏Šฉ2๐‘ฅ=+โˆž, lim๏—โ†’โˆž๏Šจ4๐‘ฅ=+โˆž, and lim๏—โ†’โˆž3๐‘ฅ=+โˆž, but then what is โˆžโˆ’โˆžโˆ’โˆžโˆ’7?

And the answer is that we cannot decide, because โˆžโˆ’โˆž is an indeterminate form.

Of course, the algebra merely affirms the intuition that the leading term determines where the polynomial goes as ๐‘ฅ tends to infinity. The 2๐‘ฅ๏Šฉ term dominates when ๐‘ฅ is large.

Example 2: Finding the Limit of a Rational Function at Infinity

Find lim๏—โ†’โˆž๏Šจโˆ’5๐‘ฅโˆ’9โˆ’2๐‘ฅ+5.

Answer

Following the strategy for polynomials, factor out leading terms first, and then cancel those: โˆ’5๐‘ฅโˆ’9โˆ’2๐‘ฅ+5=(๐‘ฅ)๏€ปโˆ’5โˆ’๏‡(๐‘ฅ)๏€ปโˆ’2+๏‡=(๐‘ฅ)๏€ปโˆ’5โˆ’๏‡(๐‘ฅ)๏€ปโˆ’2+๏‡=โˆ’5โˆ’(๐‘ฅ)๏€ปโˆ’2+๏‡.๏Šจ๏Šฏ๏—๏Šจ๏Šซ๏—๏Šฏ๏—๏Šจ๏Šซ๏—๏Šฏ๏—๏Šซ๏—๏Žก๏Žก๏Žก

Now we can โ€œsubstituteโ€ for the limits: limlimlimlimlim๏—โ†’โˆž๏Šจ๏—โ†’โˆž๏Šฏ๏—๏Šซ๏—๏—โ†’โˆž๏—โ†’โˆž๏—โ†’โˆž๏Šจโˆ’5๐‘ฅโˆ’9โˆ’2๐‘ฅ+5=โˆ’5โˆ’๐‘ฅ๏€ปโˆ’2+๏‡=1๐‘ฅร—๏€ผโˆ’5โˆ’9๐‘ฅ๏ˆ๏€ผโˆ’2+5๐‘ฅ๏ˆ=0ร—โˆ’5โˆ’2=0.๏Žก

Next is an alternative strategy.

Example 3: Finding the Limit of a Rational Function at Infinity

Find lim๏—โ†’โˆž๏Šช๏Šฉ๏Šจ๏Šฉ๏Šจโˆ’๐‘ฅโˆ’7๐‘ฅ+3๐‘ฅ+7๐‘ฅ+4โˆ’8๐‘ฅโˆ’6๐‘ฅโˆ’6๐‘ฅ+4.

Answer

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 ๐‘ฅ๏Šช), therefore the limit is โˆ’โˆž.

And the following is another way.

Example 4: Finding the Limit of a Rational Function at Infinity

Find lim๏—โ†’โˆž๏Šช๏Šฉ๏Šจ๏Šช๏Šฉ๏Šจ3๐‘ฅโˆ’๐‘ฅโˆ’๐‘ฅ+3๐‘ฅ+2๐‘ฅโˆ’8๐‘ฅโˆ’๐‘ฅโˆ’5๐‘ฅโˆ’8.

Answer

If ๐‘›โˆˆโ„๏Šฐ, and ๐‘Ž is a constant, then lim๏—โ†’โˆž๏Š๐‘Ž๐‘ฅ=0. Also, if ๐‘ is a constant, then lim๏—โ†’โˆž๐‘=๐‘. 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 lim๏—โ†’โˆž๏Šช๏Šฉ๏Šจ๏Šช๏Šฉ๏Šจ3๐‘ฅโˆ’๐‘ฅโˆ’๐‘ฅ+3๐‘ฅ+2๐‘ฅโˆ’8๐‘ฅโˆ’๐‘ฅโˆ’5๐‘ฅโˆ’8 as follows: limlimlimlimlimlimlimlimlimlimlimlim๏—โ†’โˆž๏Šช๏Šฉ๏Šจ๏Šช๏Šฉ๏Šจ๏Šช๏Šช๏—โ†’โˆž๏Šง๏—๏Šง๏—๏Šฉ๏—๏Šจ๏—๏Šฎ๏—๏Šง๏—๏Šซ๏—๏Šฎ๏—๏—โ†’โˆž๏—โ†’โˆž๏—โ†’โˆž๏Šจ๏—โ†’โˆž๏Šฉ๏—โ†’โˆž๏Šช๏—โ†’โˆž๏—โ†’โˆž๏—โ†’โˆž๏Šจ๏—โ†’โˆž๏Šฉ๏—โ†’โˆž๏Šช3๐‘ฅโˆ’๐‘ฅโˆ’๐‘ฅ+3๐‘ฅ+2๐‘ฅโˆ’8๐‘ฅโˆ’๐‘ฅโˆ’5๐‘ฅโˆ’8รท๏€น๐‘ฅ๏…(๐‘ฅ)=3โˆ’โˆ’++1โˆ’โˆ’โˆ’โˆ’=3โˆ’1๐‘ฅโˆ’1๐‘ฅ+3๐‘ฅ+2๐‘ฅ1โˆ’8๐‘ฅโˆ’1๐‘ฅโˆ’5๐‘ฅโˆ’8๐‘ฅ=3โˆ’0โˆ’0+0+01โˆ’0โˆ’0โˆ’0โˆ’0=3.๏Žก๏Žข๏Žฃ๏Žก๏Žข๏Žฃ

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