In this article, we’ll be talking about that dreaded A-word, asymptote. In my experience, students often get hung up on the term and may believe these kinds of problems are impossible. But with a solid understanding of the concepts, and a few algebraic techniques in your toolbox, it is not too difficult to locate the vertical asymptotes of a function.
The Types of Asymptotes
There are three types of asymptote: horiztonal, vertical, and oblique. This article focuses on the vertical asymptotes. Horiztonal asymptotes are discussed elsewhere, and oblique asymptotes are rare to see on the AP Exam (For more information about oblique, or slant asymptotes, see this article and this helpful video).
A vertical asymptote (or VA for short) for a function is a vertical line x = k showing where a function f(x) becomes unbounded. In other words, the y values of the function get arbitrarily large in the positive sense (y→ ∞) or negative sense (y→ -∞) as x approaches k, either from the left or from the right.
A vertical asymptote is like a “brick wall” that the function cannot cross. Imagine that you are flying in an airplane and up ahead you see a huge mountain. If you can’t go left or right around the mountain what would you do? You’d probably fly upward to avoid hitting it. Now imagine that mountain is vertical and infinitely high. Then you might fly upwards forever to avoid hitting it, and still never get over the mountain!
A function may have any number of vertical asymptotes, or none at all. Some functions even have infinitely many VAs. The graph shown below has vertical asymptotes at x = -3 and x = 1.
Because the definition involves variables approaching fixed values, it should come as no shock that limits must be involved somehow. The precise definition for a vertical asymptote goes as follows. We say that x = k is a VA for a function f(x) if either the left-hand or right-hand limit to x = k is infinite:
Finding Vertical Asymptotes
There are two main ways to find vertical asymptotes for problems on the AP Calculus AB exam, graphically (from the graph itself) and analytically (from the equation for a function). We’ll talk about both.
Determining Vertical Asymptotes from the Graph
If a graph is given, then look for any breaks in the graph. If it appears that a branch of the function turns toward the vertical, then you’re probably looking at a VA. It helps to sketch a vertical line at the x-value where you think the asymptote should be (see the graph shown above). Note, if part of the graph actually touches your vertical line, then that line is not an asymptote after all.
Determining Vertical Asymptotes from the Equation
If you need to find vertical asymptotes on the AP Exam, you will most likely not be given the graph. So you’ll need to know what to look for in the equation of the function itself. Ask yourself, where does this function have an infinite limit? We’ll see how this applies to two different kinds of functions, rational functions and trigonometric functions.
Vertical Asymptotes in Rational Functions
If your function is rational, that is, if f(x) has the form of a fraction, f(x) = p(x) / q(x), in which both p(x) and q(x) are polynomials, then we follow these two steps:
1. Factor both the numerator (top) and denominator (bottom). This is very important because if any factors end up canceling, then they would not contribute any vertical asymptotes.
2. Once your rational function is completely reduced, look at the factors in the denominator. If there is a factor involving (x – a), then x = a is a VA. If there is a factor involving (x + a), then x = –a is a VA. Note how the sign seems to be opposite both times (just like solving a factored polynomial that has been set equal to zero).
Practice Finding Vertical Asymptotes
Let’s see how our method works. Find the vertical asymptote(s) of each function.
(a) First factor and cancel.
Since the factor x – 5 canceled, it does not contribute to the final answer. Only x + 5 is left on the bottom, which means that there is a single VA at x = -5.
(b) This time there are no cancellations after factoring.
We find two vertical asymptotes, x = 0 and x = -2.
Vertical Asymptotes for Trigonometric Functions
The method of factoring only applies to rational functions. However, many other types of functions have vertical asymptotes. Perhaps the most important examples are the trigonometric functions. Out of the six standard trig functions, four of them have vertical asymptotes: tan x, cot x, sec x, and csc x. In fact, each of these four functions have infinitely many of them!
For example, f(x) = cot x has a VA at every integer multiple of π. In other words, x = n π is a VA for every n = 0, ±1, ±2, ±3, …
Using your Graphing Calculator
More general functions may be harder to crack. If you are working on a section of the exam that allows a graphing calculator, then you may simply graph the function and try to spot the breaks in the graph at which the y-values become unbounded. Some calculators, like the TI-84, even have an option called detect asymptotes, which will automatically graph the VAs. Just be careful, though; if your viewing window is too small, then you may miss a VA.
Asymptotes are just certain lines that tell us about the behavior of functions. A vertical asymptote shows where the function has an infinite limit (unbounded y-values). It is important to be able to spot the VAs on a given graph as well as to find them analytically from the equation of the function. Your graphing calculator can also help out. With a little time and practice, these techniques can easily be mastered, and so vertical asymptotes don’t have to be the “brick wall” that stops you from going far on the AP Calculus exam!
About Shaun Ault
Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!
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“An Inconvenient Newspaper” is an essay about the recent closure of the Buenos Aires Herald, a paper that wrote against the Argentine military dictatorship, in English, in the 1970s and 1980s. The Buenos Aires Herald closed in July, just as an Argentine indigenous rights activist disappeared. The full profile of Robert Cox, the director of the Herald, was published in a Portuguese translation in issue no. 133 of the Brazilian magazine Piauí, released in October 2017. This English translation is an abridged version of the original Spanish article by Josefina Licitra.
“Any news?” That’s how Robert Cox greets me. He says “hello” and “nice to meet you” with an affectionate kiss on the cheek. But in the following sentence he always probes for the unexpected, for the possibility of news. It’s 10 a.m. on a Thursday and Cox looks like he just woke up. His eyes are still sleepy and his white hair finger-combed.
“Not that I know of,” I reply.
Cox makes coffee in the kitchen and brings it to the living room: a pleasant space scattered with paintings, family photos, and other decorations. He lives with his wife, Maud Daverio, in Charleston—in the United States—but also keeps this old, elegant Buenos Aires apartment, which he visits every year. This is where he lived after getting married, in 1961. This is where his five children were born. This is where he lived when the Buenos Aires Herald, the English-language newspaper that he directed from 1968 to 1979—one of a kind in Latin America—became the Argentinian publication that spoke out about human rights violations during the last military dictatorship, at a time when no other media institution would. And this is the place that he had to leave when a series of threats—also directed against his wife and one of his children—forced his family into exile.
Cox looks through the voile curtains. Outside the window is a narrow street lined with the pompous buildings of the Recoleta neighborhood, one of the most European areas of Buenos Aires. “I don’t know what happened with Santiago Maldonado…” he says, and clicks his tongue with an audible tsk. “Still no news? Weird.”
Santiago Maldonado is—was?—an artisan who supported the struggle of radical indigenous groups that reclaim land in Patagonia. This past August 1st, after a protest that stopped traffic, he disappeared in the middle of a confrontation with the Gendarmería—border officers. Some say that the police arrested him and accidentally killed him through the use of excessive force. Others say that there is no evidence to show that the government was at fault—and to this day there still isn’t—but they also can’t come up with a different explanation for his disappearance. Since then, demands to find Maldonado alive—or to find him at all—have deepened the divisions between Argentina’s governing party and its opposition. While the government refers to Maldonado as an “artisan,” kirchneristas and left-wing parties call him a desaparecido—one of the “disappeared.”
That term, in Argentina, dredges up the history in which Robert Cox was involved.