Estimating Limits from Graphs (Topics 1.2-1.3)

Unit 1 - Day 3

​Learning Objectives​
  • Use graphical representations to estimate limits, including one-sided limits. 

  • Explain why a limit may not exist at a specific x-value: function values are unbounded, a function oscillates near an x-value, or the limit from the left does not equal the limit from the right

​Success Criteria
  • I can explain the idea of a limit.

  • I can use proper limit notation.

  • I can use tables and graphs to estimate limits.

Quick Lesson Plan
Activity: Quiz, Quiz, Trade

Additional Materials

  • Sticky Notes


Set of 32 Limit Cards


Each student is given a card with a unique graph. Without help from others, the student re-writes the limit expression on their sticky note and evaluates the limit. The sticky note is then attached to the back of the card. When directed, students form pairs and show each other the fronts of their cards. Each partner takes turns reading the limit expression of their partner’s card out loud and evaluating it using the graph. The person holding the card then checks their partner’s answer against the sticky note. When both parties agree on the correct limits, students exchange cards and seek a different partner. If changes are needed, students make corrections to their original answer before exchanging cards and finding a new partner. This process repeats several times. At the end of the activity, students should retrieve their original notecard and see if corrections or changes have been made.

Teaching Tips

The teacher should have a couple extra completed cards in their hand to work with students who may be without a partner at any particular moment so that students are always discussing a graph. While students are working, the teacher is monitoring student communication for precise vocabulary. This is a great way to engage kinesthetic learners and require student-to-student conversations.

Exam Insights

Interpreting graphs is a required skill for both the multiple choice and free response sections of the AP Exam. This activity is a comfortable way for students to practice this skill and learn from their peers.

Student Misconceptions

Students need to see any changes (correct or not!) to their original solution for two reasons: to resolve any of their original misconceptions and to ask about potential incorrect answers that may have appeared through the course of the activity. This reflection is a vital part of the learning process.

Suggested Homework

Include several problems of the form f(x) = |x + a| / (x + a). Students should recognize this as a horizontal shift of f(x) = |x| / x and ultimately be able to visualize the graph for future assignments or assessments.

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