The Limit of a Function

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Objective

SWBAT evaluate the limit of a function using a graph or a table.

Big Idea

Use many different ways to find the limit of a function.

Launch

10 minutes

We studied limits of sequences earlier in the year, but today we are going to extend that knowledge and think about limits of functions. The skill set is definitely similar, but adding a graphical approach will enhance student understanding.

I start the lesson by activating the prior knowledge of a limit of a sequence by having students work in their table groups on questions #1 from this worksheet. We will share our responses after students get their answers.

At this point I want to transition to the limit of a function, so I will explain to students that we are going to be finding the limits of graphs and equations instead of sequences. Next I will define the limit of a function and introduce the notation (including notation for right side and left side limits). I make a big deal about the fact that the limit must approach the same value from both sides in order to exist.

Explore

15 minutes

After the initial phase of defining the limit, I try not to give too much instruction and just have students go at questions #2 and #3 on the worksheet. They are using graphs and equations to find limits, but I do not give them any method to find the limit - I want to see where their intuition takes them.

As students are working I will ask them questions to get a feel for the reasons that they are giving for the limits they choose. I am also making mental notes about the techniques that are most prevalent. Below are a list of common misconceptions that I see when working on this lesson. Hopefully you can use this list to anticipate the misconceptions and be able to guide them to better thinking.

  1. If a graph shows that a function is undefined at a certain point, students will automatically say that there is not limit even if the function approaches the same value from both sides. This is true for equations too; if they test the point and it is undefined, they will automatically say there is no limit.
  2. Students have a tough time when asked to find the limit at a point where the graph is continuous (such as #2b on the worksheet). They are usually over-thinking things and have a tough time seeing that the limit is just the value of the function at that point.
  3. Limits to positive or negative infinity can be tough. Sometimes students will say that the limit is infinity, even if the function approaches a value. Many times the student incorrectly focuses on what the x-value is approaching instead of what the y-value is approaching.

 

Share and Summarize

20 minutes

When it is time to share the answers for #2 on the worksheet, I will have students share their answers with the class and give their reasons. I will usually randomly select students and hope that a few of them got the wrong answers - the discussions about the validity of wrong answers are usually very beneficial to the class. I talk more about this in the video below.

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For questions #3, I make sure to get a variety of different methods by selecting the students to share their thinking. Here are some different methods I try to elicit:

  • Thinking of what the graph looks like - many students will visualize the shape of the graph and think about undefined values, gaps, and asymptotes.
  • Actually graphing the function - some students will actually graph the function on their graphing calculator and inspect it. Problems may arise if the calculator does not show discontinuities or the resolution is too low to get an exact answer.
  • Plugging in values - many students will plug in x-values manually or use the table on their graphing calculator to see what the function approaches.
  • Inspecting the equation - For a function like y= (x^2 + 8x)/x, a student may factor the numerator and cancel common factors to get a form they can work with.

Next, for question #4 I will go over the nuts and bolts of using a table on the graphing calculator to evaluate the limit. Many students will need assistance setting up the table and changing the setting so that they can enter in the x-values instead of it being automatic.

To close the lesson, I ask students about the similarities and differences in finding the limit of a sequence and the limit of a function. I want to reinforce the idea that in both cases we are looking at what value is being approached, but that the representations are different. Finally, I will assign some limit problems from their textbook as homework.