Vertical Asymptotes in Rational Functions 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. So you’ll need to know what to look for in the equation of the function itself. If you need to find vertical asymptotes on the AP Exam, you will most likely not be given the graph. Determining Vertical Asymptotes from the Equation Note, if part of the graph actually touches your vertical line, then that line is not an asymptote after all. It helps to sketch a vertical line at the x-value where you think the asymptote should be (see the graph shown above). If it appears that a branch of the function turns toward the vertical, then you’re probably looking at a VA. If a graph is given, then look for any breaks in the graph. Determining Vertical Asymptotes from the Graph 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 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: The precise definition for a vertical asymptote goes as follows. The graph shown below has vertical asymptotes at x = -3 and x = 1.īecause the definition involves variables approaching fixed values, it should come as no shock that limits must be involved somehow. Some functions even have infinitely many VAs. Then you might fly upwards forever to avoid hitting it, and still never get over the mountain!Ī function may have any number of vertical asymptotes, or none at all. Now imagine that mountain is vertical and infinitely high. If you can’t go left or right around the mountain what would you do? You’d probably fly upward to avoid hitting it. Imagine that you are flying in an airplane and up ahead you see a huge mountain. 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.Ī vertical asymptote is like a “brick wall” that the function cannot cross. Vertical AsymptotesĪ vertical asymptote (or VA for short) for a function is a vertical line x = k showing where a function f(x) becomes unbounded. 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). This article focuses on the vertical asymptotes. There are three types of asymptote: horiztonal, vertical, and oblique.
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.
In my experience, students often get hung up on the term and may believe these kinds of problems are impossible. In this article, we’ll be talking about that dreaded A-word, asymptote.
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