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NYU Steinhardt EMAT Teaching Residency
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This is a pre-calculus introduction to limits




12, 11  

Title – Precalculus: Introduction to Limits
By – Ann Morris
Primary Subject – Math
Grade Level – 11-12


    By the end of this lesson, the student will be able to compute a basic limit of a function using limit notation.

Pre-Class Assignment:

    Students will create a poster titled “There is a Limit to…..” or “There is No Limit to….”. They will draw or cut out pictures from magazines. I always show them several posters created in classes before them or posters that I have created. For example: “There is No Limit to the Number of Pairs of Shoes One Can Have” or “There is a Limit to the Amount you can Spend on a Credit Card”. (These posters will be hung up in the room or in the hall.)

Resources: Precalculus Textbook

Time Required: 90 minutes


    Overhead projector or Interwrite Pad; graphics calculator (TI-84 recommended) and some method of posting a graph on a large screen (TI-Presenter/TI-84 calculator, Epson projector/document camera, etc.)


      Introduce the lesson by having students show their posters. Discuss how there is or is not a limit to the topic they picked.
      Discuss how many mathematical problems involve the behavior of a function at a particular number. Present the students with the question: “What is the value of the function f(x) when x = c?” Make sure the students understand the concept by telling them that they are to find where “y” is when “x” is at a particular number. Then start explaining that the idea of


      is the behavior of a function (y-value) when the x-value is getting close to or near a particular number, rather than when x = a particular number.
      Introduce the word


      . Start with an example such as:


        Walk to a particular desk and stand close to it; then go back and walk to the same desk and sit down in it. Ask students what the difference is in those 2 different actions.
        Another example would be to point out on the x-axis of getting close to a number and comparing that to actually getting to the number.
        It is very important that students understand the difference between the terms approaching and actually getting there.


      Give them the following examples and let them graph them on a graphics calculator:



y=x 2

        Ask them to find what x is getting close to when y=4.

y=x 2 -x+2

        Ask them to investigate the behavior for values of y as x nears 2 from the left. Point out that as ‘x’ gets closer to 2, ‘y’ gets closer to 4. Make a

Table of Values

        like the following:
x f(x)
1.0 2.000000
1.5 2.750000
1.8 3.440000
1.9 3.710000
1.95 3.852500
1.995 3.985025
1.999 3.997001


        Then make a chart as ‘x’ approaches 2 from the right. Point out that as ‘x’ gets closer to 2 from, ‘y’ again gets closer to 4.
x f(x)
3.0 8.000000
2.5 5.750000
2.2 4.640000
2.05 4.152000
2.01 4.030100
2.005 4.015025
2.001 4.003001



y = sin x

        Discuss that as x is getting closer to pi that y is approaching 0.
      Then give them the formal definition of limit from the textbook.

Calculator Exploration:

      Have students create a Table of Values for functions such as:

y = sin x/x
y= (.1x 4 -.8x 3 -2x)/(x-4)

      and discuss that is happening to the y-value when x is getting closer to 0.

      Introduce the notation for limit as x is approaching a specific number. You can also re-introduce the

infinity symbol

    and discuss what is happening to a specific function as ‘x approaches infinity’. Demonstrate how to write the limit notation as x is approaching infinity.

Cooperative Learning Activity:

    Number the students from 1-7, put them in their respective groups and give them several problems to work from the textbook using the limit notation. Then have each group to come up with their own examples to be presented tomorrow at the beginning of class.


    Make an assignment from the textbook.

Prerequisite to:

    Nonexistence of Limits (oscillating functions, etc)

E-Mail Ann Morris !

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