Asymptotes, Limits at Infinity, and Continuity in Maple (Classic Version for Windows)

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1 Asymptotes, Limits at Infinity, and Continuity in Maple (Classic Version for Windows) Author: Barbara Forrest Contact: Copyrighted/NOT FOR RESALE version. Contents Objectives for this Lab i 2 Using Maple s plot options for find its i 3 Practice Exercises v Objectives for this Lab Understand and use the options for Maple s it command. Use plots to visually verify calculated its in Maple. Explore asymptotes, its at infinity, and continuity using Maple. 2 Using Maple s plot options for find its You have already seen how to use Maple s it command to find the it of functions in one variable. In this lab you will look at the options available for the it command. Recall that the general syntax for finding the it a function of one variable x a f(x) with options in Maple is it( function of x, x = it point, options ); To illustrate, let s do an example. Exercise: Calculate x x. Open a blank worksheet in Maple CLASSIC version. Clear Maple s memory by entering: i

2 Define the function f(x) = x [> f := x -> /(-x); What does the graph of f(x) = x by entering the following: look like? [> plot(f(x), x=-0..0, y = -0..0); Notice that the graph looks very similar to the hyperbola, f(x) = /x, except that it is shifted to the right by unit and then reflected through the x-axis? Question: Can you explain why this is the case using your knowledge about transformations of functions? Also notice that Maple plots a vertical line at x =. There is really a break in the graph at this point (called a discontinuity). In fact, a vertical asymptote occurs here. (Note: A vertical asymptote occurs at any value that causes the denominator of a function to be zero with the numerator non-zero.) Maple needs a little help in the case where a function has a discontinuity. If you ever plot a function and a vertical line such as this appears, use Maple s discont=true option. This tells Maple to look for discontinuities when plotting the given function: [> plot(f(x), x=-0..0, y = -0..0, discont = true); That s better! Now suppose we want to know x x. From the graph, we can see that as x, f(x) 0. (As x gets larger and larger, the graph of f(x) gets closer and closer to the horizontal axis x = 0.) In Maple, enter the following command to confirm our observation: [> it(f(x), x=infinity); Indeed, Maple confirms our suspicion that the it of the function as x is 0. In other words, x x = 0. What about x x? From the plot, we can see that at x =, there is a break in the graph called a discontinuity since a vertical asymptote occurs there. From this observation, it seems that the it of this function as x does not exist since one part of the graph goes upward and the other part of the graph goes downward. In Maple, enter the following command to confirm our observation: [> it(f(x), x=); Maple returns undef ined This is Maple s way of telling us that the it does not exist. ii

3 Exercise: Calculate x 2 x x. Clear Maple s memory by entering: Define the function f(x) = x2 x by entering the following: [> f := x -> abs(x 2 -)/(x-); What does the graph of f(x) = x2 x look like? [> plot(f(x), x=-0..0, y = -0..0); Notice that there is a discontinuity (vertical line on the graph). Thus, we should use the discont = true option in Maple s plot command. [> plot(f(x), x=-0..0, y = -0..0, discont=true); QUESTION: What is x 2 x x? In Calculus class, there is a theorem that states A two-sided it exists if and only if both of the one-sided its EXIST and are EQUAL. We can use Maple to verify this theorem for the function f(x) = x2 x. To find the it of a function from the left in Maple you need to use the left direction option: [> it(f(x), x=, left); f(x) x a Maple tells us that the function f(x) goes toward 2 as you approach x = from the left. Visually confirm this is the case on the graph of f(x). To find the it of a function from the right in Maple you need to use the right direction option: [> it(f(x), x=, right); f(x) x a + Maple tells us that the function f(x) goes toward 2 as you approach x = from the right. Since the left-hand it is 2 and the right-hand it is 2, the two one-sided its are not equal. Thus, by the theorem stated above, we can say that the (two-sided) it does NOT exist. In fact, let s check the two-sided it using Maple: [> it(f(x), x=); iii

4 Maple returns undefined which confirms that the it does not exist. Exercise: Let f(x) = x2 2x x 2 x. Calculate f(x) and f(x). x Clear Maple s memory by entering: Define the function f(x) = x2 2x x 2 x [> f := x -> (x 2-2*x)/(x 2 - x); What does the graph of f(x) = x2 2x x 2 x [> plot(f(x), x=-5..5); by entering the following: look like? Notice that there is a discontinuity (vertical line on the graph). Thus, we should use the discont = true option in Maple s plot command. [> plot(f(x), x=-5..5, discont=true); QUESTION: What is x x 2 2x x 2 x? [> it(f(x), x=); Maple returns undefined which confirms that the it does not exist since there is a vertical asymptote at x =. QUESTION: What is x 2 2x x 2 x? In answering this question, you should notice that the function is not defined when x = 0. This suggests that there may be a removable discontinuity at x = 0. Let s use Maple to check for a removable discontinuity by entering: [> plot(f(x), x=-5..5, discont=[showremovable]); x Indeed, Maple displays a hole at x = 0. However, to answer the original question 2 2x x 2 x [> it(f(x), x=0); Maple tells us that x 2 2x x 2 x = 2. you can enter You should confirm that the left-hand it and the right-hand it as x approaches 0 are both equal to 2 and so by the Theorem on the previous page, even though there is a removable discontinuity at x = 0 the it still exists and is equal to 2. Exercise: Let f(x) = sin( x ). Calculate f(x). Clear Maple s memory by entering: Define the function f(x) = sin( x ) by entering the following: [> f := x -> sin(/x); What does the graph of f(x) = sin( x ) look like? [> plot(f(x), x=-5..5); iv

5 There are a few things to notice in the plot. First, the graph of f(x) always lies between y = and y =. Next, there seems to be something special occuring near x = 0. Let s zoom in by changing the x interval in the plot command: [> plot(f(x), x= ); Let s look at an even smaller interval: [> plot(f(x), x= ); NOTICE: The graph of f(x) always lies between y = and y = but it oscillates wildly near x = 0! The function f(x) = sin( x ) is an example of an oscillatory discontinuity. 3 Practice Exercises Complete the following exercises in a Maple worksheet and write your answers in the space provided (where applicable). Save the worksheet as asymptotes.mws. Questions from these practice exercises may appear on your course assignments.. Create the following text input. At the top of a new Maple worksheet, type the following lines in a text region. Maple Lab : Asymptotes, Limits at Infinity, and Continuity Name: ID: Date: Type your name here. Type your ID number here. Type today s date here. You can enhance the fonts in any way you wish (e.g., change the font, fontsize, bold, etc.) Insert an execution group after the last line of the text region. Enter Maple s restart: command. 2. Calculating its. (a) In Maple, create a text region and enter the following sentence: Finding two-sided its. Insert an execution group (command prompt) after the the text region. (b) In Maple, create the function f(x) = sin( x ) x. HINT: Use f := x -> sin(abs(x))/x; (c) Plot f(x) with the x-axis varying from 0 to 0. Notice that there seems to be a vertical line (jump) between y = and y =. Fix the plot by including discont = true in the plot command. (d) QUESTION: Considering the graph, GUESS the value of f(x)? (e) Calculate the left-hand it using Maple. (f) Calculate the right-hand it of f(x) using Maple. f(x)? f(x)? + v

6 (g) QUESTION: Does f(x) exist? Explain your answer. (Hint: Apply the Theorem from this lab.) (h) Confirm your answer by calculating [> it(f(x), x=0); 3. Save the worksheet. Save the worksheet as asymptotes.mws. You may print the worksheet if you would like a copy for your notes. To print a copy of the worksheet, from the File menu, choose Print and click OK. vi