MA 223 Lecture 26 - Behavior Around Vertical Asymptotes Monday, April 9, 208 Objectives: Explore middle behavior around vertical asymptotes. Vertical Asymptotes We generally see vertical asymptotes in the graph of a function when we divide by zero. For example, in the function () f(x) = x, f is undefined at x = 0, and for x s very close to zero, the function values are very large. Recall that we use a + or as a superscript to indicate its from the right or left, and in this case we have the it from the right being (2) x 0 + x =, and from the left (3) x 0 x =, since divided by a very small negative number is a very large negative number. The graph looks like this. Other shapes. The most noticeable difference in the look around a vertical asymptote is whether the graph goes up or down on either side of the asymptote. With an even power, the graph will go in the same direction on either side, like for (4) f(x) = x 2, we have that (5) x 0 x 2 = x 0 x 2 = x 0 + x 2 =.
MA 223 Lecture 26 - Behavior Around Vertical Asymptotes 2 Multiplying by constants will stretch or compress the graph vertically, and if the constant is negative, flip things upside-down. So, for example, the function (6) f(x) = 2 x looks like this. Vertical Asymptotes in a Rational Function The basic reasoning about vertical asymptotes in general goes something like this. Consider the function (7) f(x) = (x + 3)(x + ) (x 2) 3. The first thing to notice is that f is undefined at x = 2. Using x = 2 + to mean a number a little bit bigger than 2, we have (8) f(2 + ) = (2+ + 3)(2 + + ) (2 + 2) 3 = 5+ 3 + (0 + ) 3 5 + 0 +. So basically, we have something close to 5 divided by something very small and positive, and so f(2 + ) is really big and positive. Looking at x = 2, we have (9) f(2 ) = (2 + 3)(2 + ) (2 2) 3 = 5 3 (0 ) 3 = 5 0. The numerator is still close to 5, and the denominator is still very small, but now the denominator is negative. Therefore, f(2 ) is really big and negative. Limits from the left or right mostly matter, when we re multiplying or dividing by zero. The its are from the right (0) x 2 + (x + 3)(x + ) (x 2) 3 =,
MA 223 Lecture 26 - Behavior Around Vertical Asymptotes 3 and from the left () x 2 (x + 3)(x + ) (x 2) 3 =. The graph looks like this (pretending that Maple is actually drawing an asymptote). Behavior Around Asymptotes OK. So let s suppose we have a rational function, and have both the numerator and denominator factored, like (2) f(x) = (x + 2)(x 3)2 (x + ) 2 (x 4). We can easily see where the asymptotes are, at x = and x = 4 (in fact, those are the equations for the asymptotes). Let s look at x = first. On some level, we re going to consider the factors in two groups, like (3) f(x) = Around x =, f is roughly going to be (4) (x + 2)(x 3)2 x 4 ()( 4) 2 ( 5) Therefore, the it from the left and right will be. (5) x f(x) = x + f(x) = Around x = 4, we want to separate the factors like (6) f(x) = So, around x = 4, f is roughly (x + ) 2. 0 = 4 2 5 0 2. f(x) =. x (x + 2)(x 3)2 (x + ) 2 (x 4). 6 2 2 (7) 5 2 0 = 24 25 0. If x is a little less than 4, then f is negative, so (8) f(x) =. x 4 If x is a little bigger than 4, then f is positive, so (9) x 4 + f(x) = +.
MA 223 Lecture 26 - Behavior Around Vertical Asymptotes 4 Basic Principle. A factor in the denominator will correspond to an asymptote where it is equal to zero. The other factors in the function multiplied and divided together will be approximately equal to some number, and the main thing to notice is whether that number is positive or negative. You then want to determine the behavior around the asymptote based on x s a little smaller and a little larger than the zero. Here are a couple more examples. Example. Consider the function (20) f(x) = (x + 3)2 (x ) (x + 2) 3 x 2 (x 2). Around x = 2, the function looks like (2) 2 ( 3) ( 2) 2 ( 4) 0 3. The other stuff is positive, so the behavior around this asymptote will act like x 3, and (22) x 2 f(x) = and Around x = 0, the function looks like (23) 3 2 ( ) 2 3 ( 2) 0 2. x 2 + f(x) = +. The other stuff is positive, so the behavior around this asymptote will be like x 2, and (24) x 0 f(x) = + and Around x = 2, the function looks like 5 2 () (25) 4 3 2 2 0. The other stuff is positive, so the behavior will be like x, and (26) x 2 f(x) = and Example 2. Consider the function f(x) = +. x 0 + f(x) = +. x 2 + (27) f(x) = (x + 2)3 (x ) (x + 3) 2 x 3 (x 2) 2. Around x = 3, the function looks like (28) ( ) 3 ( 4) ( 3) 3 ( 5) 2 0. 2 The other stuff is negative, so the behavior around this asymptote will act like x 2, and (29) x 3 f(x) = and Around x = 0, the function looks like (30) 2 3 ( ) 3 2 ( 2) 2 0. 3 f(x) =. x 3 + The other stuff is negative, so the behavior around this asymptote will be like x 3, and (3) f(x) = + and x 0 f(x) =. x 0 + Around x = 2, the function looks like (32) 4 3 () 5 2 2 3 0 2.
MA 223 Lecture 26 - Behavior Around Vertical Asymptotes 5 The other stuff is positive, so the behavior will be like x 2, and (33) x 2 f(x) = + and Consider the function (34) f(x) =. Where are the vertical asymptotes? Quiz 26 x 2 + f(x) = +. x 3 (x + 2) 3 (x 2) 2. 2. Find the its from the left and right at each of the vertical asymptotes. Homework 26 For each of the following functions, find all of the vertical asymptotes, and the its from the left and right for each.. f(x) = 2. f(x) = 3. f(x) = 4. f(x) = 5. f(x) = x 2 (x + 2) 2 (x 2) 3 (x + 2)2 x 2 (x 2) 3 (x + 2)3 x 3 (x 2) 2 (x + 2) x 3 (x 2) 3 x(x 3) 3 (x + 2) 2 (x 2) 3 Answers: ) x = 2: From left, and from right. x = 2: From left, and from right +. 3) x = 0: From left, and from right +. x = 2: From left +, and from right +. 5) x = 2: From left, and from right. x = 2: From left +, and from right.