C.6 Limits of Rational Functions

Let's now examine the limit as x goes to positive or negative infinity of rational functions, which are formed by dividing one polynomial by another polynomial. We'll make direct use of the ideas of dominance that we developed on the preceding screen.

Indeed, building off of that earlier work, the reasoning here is straightforward. Let's consider a general rational function of the form (𝐴𝑥𝑁+ (smaller terms))(𝐵𝑥𝑀+ (smaller terms)).

Quick, informal reasoning: Identify the term in the numerator with the highest power, and the term in the denominator with the highest power. You can then ignore all of the smaller terms just as we did for the polynomials earlier. We can do this both as 𝑥 →∞,

and 𝑥 → −∞,

Then compare the power in the numerator to the power in the denominator. There are three possibilities, each with its own result: (1) the highest power in the numerator is the same as that in the denominator (𝑁 =𝑀); (2) the highest power in the denominator is greater than that in the numerator (𝑀 >𝑁);) (3) the highest power in the numerator is greater that in the denominator (𝑁 >𝑀).

We'll examine an example of each below to determine the various results.

Formal development of the limit: If you need to develop the limit more formally, divide every term in the function by the largest power in the denominator and then proceed to find the limit.

Let's consider an example of each possibility at both ∞ and −∞ to determine the various results.

Case 1: Rational Function with equal highest powers in the numerator and denominator

Let's look first at what happens when the largest powers in the numerator and denominator are equal, and as 𝑥 → +∞.

Let's next consider a different rational function with equal largest-powers in the numerator and denominator, now as 𝑥 → −∞.

Summary of Case 1, and Horizontal Asymptotes

The explorations above show that when the largest powers in the numerator and denominator are equal, then as 𝑥 →∞ lim𝑥→∞(𝐴𝑥𝑁+ (smaller terms))(𝐵𝑥𝑁+ (smaller terms))=lim𝑥→∞𝐴𝑥𝑁𝐵𝑥𝑁=𝐴𝐵 That is, the limit is the ratio of the coefficients of those largest-power terms.

y = A/B is a horizontal asymptote

Furthermore, because the limit is a constant, as 𝑥 →∞ the function resembles the horizontal line 𝑦 =𝐴𝐵, which is known as a horizontal asymptote.

Similarly, as 𝑥 → −∞ lim𝑥→−∞(𝐴𝑥𝑁+ (smaller terms))(𝐵𝑥𝑁+ (smaller terms))=lim𝑥→−∞𝐴𝑥𝑁𝐵𝑥𝑁=𝐴𝐵 And now as 𝑥  → −∞ the function again resembles the horizontal line 𝑦 =𝐴𝐵, and so there is a horizontal asymptote in the negative direction as well.

We'll provide some practice problems below that ask you to determine the horizontal asymptote for a function, which means finding the limit as 𝑥 →∞ and 𝑥 → −∞ as we did above.

For now, let's move on to Case 2.

Case 2: Rational Function where the denominator has the largest power

Summary for Case 2: If the largest power in the denominator of a rational function is larger than any power in the numerator, then the limit at ±∞ is zero. The reason is simple: the denominator is growing faster than the numerator, and so dominates, or "wins," the overall behavior of the fraction, forcing the function to zero as x grows large in either direction.

Case 3: Rational Function where the numerator has the largest power

The result here won't be surprising, but let's take a look at an example function anyway for completeness.

Summary for Case 3: If the largest power in the numerator is greater than any power in the denominator, then the limit as 𝑥 →∞ and the limit as 𝑥 → −∞ does not exist, and is either ∞ or −∞.

Most students find trying to memorize the results above . . . well, kinda overwhelming. So don't. Instead, use the reasoning skills we've been building on this and the preceding screens, and write down a line for yourself that shows the dominance: as 𝑥 → ±∞, only the largest terms in the numerator and denominator matter. Once you do that, the resulting conclusion will be clear depending on which, if either, of those terms is larger, And if they're equal, the resulting fraction that is the limit (and horizontal asymptote) presents itself.

You can practice this simple move in the problems below, along with writing out the formal reasoning if your course requires that.

[That all said, you'll find a graphic that summarizes all of the results of this and the preceding screen in "The Upshot" at the bottom of this screen.]

Practice Problems: Limits of Rational Functions

Practice Problem 4

Find the horizontal asymptotes of  5𝑥2+𝑥−33𝑥2−2𝑥+5.

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Asking for the horizontal asymptotes is simply another way of asking what the function does as 𝑥 →∞ and 𝑥 → −∞. Your intuition is probably that as 𝑥 gets very large (in the positive or negative direction), the 𝑥2 terms in the numerator and denominator dominate, and the function becomes essentially 5𝑥23𝑥2, and hence the asymptote is 𝑦 =53. This is correct. To show that this is the case, divide each term in the fraction by the largest power in the denominator, 𝑥2: lim𝑥→∞5𝑥2+𝑥−33𝑥2−2𝑥+5=lim𝑥→∞5𝑥2𝑥2+𝑥𝑥2−3𝑥23𝑥2𝑥2−2𝑥𝑥2+5𝑥=lim𝑥→∞5+1𝑥−3𝑥23−2𝑥+5𝑥=53✓ Similarly, lim𝑥→−∞5𝑥2+𝑥−33𝑥2−2𝑥+5=lim𝑥→∞5+1𝑥−3𝑥23−2𝑥+5𝑥=53✓ Hence the horizontal asymptote is 𝑦 =5/3. See the graph.

This has been a pretty quick pass through the various possibilities for the limit as 𝑥 →∞ of rational functions.

What questions or comments do you have? Are you perhaps working on a homework problem that you're having trouble solving? Please let us know over on the Forum.

And on the next screen, we'll examine exponential and logarithmic functions and see how their dominance compares to that of polynomials. Once you have that understanding, you'll have essentially all of the tools you need to work with any limits you encounter in this first part of the course!

The Upshot

1. When considering lim𝑥→±∞, let dominance in the numerator and denominator do their work: lim𝑥→∞(𝐴𝑥𝑁+ (smaller terms))(𝐵𝑥𝑀+ (smaller terms))=lim𝑥→∞𝐴𝑥𝑁𝐵𝑥𝑀 lim𝑥→−∞(𝐴𝑥𝑁+ (smaller terms))(𝐵𝑥𝑀+ (smaller terms))=lim𝑥→−∞𝐴𝑥𝑁𝐵𝑥𝑀 Once you've done that step, the rest of the reasoning should be clear.
2. If you'd like a summary graphic of this and the preceding screen that you can save:
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