### CA I.8 Introduction to Limits

Purpose: To develop students understanding of limiting behavior numerically, graphically and algebraically.

Classroom Procedure: After an initial introduction to this activity students will work on the first two problems in small groups. There may be points in this work that you would like to call the class together to share the progress that they have made. This activity generally takes two days in class to complete and somewhere in Q3 is a good breaking point as after an initial discussion of this problem students can complete this problem at home and return ready to discuss their findings.  It is important to establish in 3d that any exponential function will eventually dominate over any power function and consider the implications of this for the limits on the previous page.  For Q4, students should work individually, then in small groups and then be ready to participate in whole class discussion. Again for Q5 it is best if students work individually, then in small groups and then as a whole class. Q6 is an important whole class discussion where the limit definition of asymptotes is established using students graphical understanding of asymptotes and the understanding of limits they have developed in this activity. Q7 is difficult for the students and is best done in class.

Ideas this Activity Builds On: This activity builds on students intuitive ideas of getting "closer and closer" to something  and "getting larger and larger" as well as their graphical understanding of asymptotes. It also uses their understanding of what  exponential functions and power functions are.

Introduction/Motivation of the Activity: This activity can be motivated through the characterization of functions. Limits are another way to characterize the behaviour of functions. In the case of rational functions they are a more interesting characterization than a characterization using rate of change.

Need to Establish by the End of Activity/Wrap-Up: Students should leave this activity understanding limits graphically, numerically and algebraically (not using an epsilon delta definition though). They should understand the limit definitions of vertical and horizontal asymptotes as well as be able to recognize asymptotes graphically. We expect that they can easily move from a limit statement to a graphical or numerical representation that behaves in the same way.

Additional Notes: There are no synthesis questions for this activity but it is important to assign homework questions over this material. We do this throughout the course via our text homework using an online homework system.

#### Teacher Journal CA I.9

• Introduction to Limits  I have found students generally enjoy this activity and find it more concrete than the work that they have just done on rate of change equations. They struggled with Q3 in general and part (d) in particular. I think because the find it hard to believe that any exponential function will dominate any power function. It is also difficult because they are asked to generalize by themselves which is always a difficult step. It is good to talk to small groups during this process and to have a whole class discussion when everyone has worked on it. If this is where you break to send them away (which is appropriate so that students have time to think about this question) it is a good idea to ask students to formulate a plan for part (d) before leaving the classroom. Students struggle again with Q7 where they are being asked to reason algebraically from a set of statements that we are taking to be true about how sums of functions behave and which types of functions dominate in the long term. I think it can be helpful to discuss this question in class and ask a number of students to explain their reasoning process.
Posted Jun 10, 2012, 7:23 PM by Paula Shorter
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