Lesson 3: 1-3 Finding Limits from Graphs
Duration of Days: 2
Lesson Objective
The student will be able to:
Estimate the value of a limit by tracing the path of a function on a graph from both the left and right sides.
Identify and distinguish between the limit value and the function value (f(c)) using open and closed circles.
Determine the existence of a limit based on the convergence (or lack thereof) of the two "branches" of a graph.
Categorize different types of graphical behaviors that lead to a limit "Does Not Exist" (DNE) status.
How do you find a one sided limit?
When does a limit fail to exist?
If there is a hole in the graph at x = c, why does the limit still exist?
What is the visual difference between a "jump" and a "removable" discontinuity in terms of limits?
Graphical Estimation
Two-Sided Limit
Removable Discontinuity (Hole)
Jump Discontinuity
Vertical Asymptote (Infinite Discontinuity)
Convergence
Divergence
F-IF.7: Graph functions expressed symbolically and show key features of the graph
Description: This lesson moves the limit concept into a purely visual arena. Students are given complex graphs with multiple "breaks." They practice the "Two-Finger Rule": placing one finger on the curve to the left of the target x-value and one to the right, sliding them toward the target. If the fingers meet at the same y-height, that height is the limit.
Purpose: This builds the visual intuition required for the formal definition of Continuity later in the unit. It trains students to look at "intent" (where the graph is going) rather than "reality" (where the dot actually is).
DOK Level 1 (Recall): Identifying the limit value for a continuous portion of a graph.
DOK Level 2 (Skill/Concept): Evaluating limits at points where there is an open circle or a displaced point.
DOK Level 3 (Strategic Thinking): Determining if a limit exists at a point where the graph exhibits different behaviors (e.g., approaching an asymptote from one side and a constant from the other).
For Struggling Learners (Scaffolding):The "Wall" Technique: Have students draw a vertical line (a wall) at x = c. They must look at where the graph "hits the wall" from the left and right.
Open vs. Closed Circles: Use a "Traffic Light" analogy: a closed circle is a stop, but an open circle is just a pothole you can drive "over" (approach).
Non-existent limits: Provide a "DNE Checklist":
1. Do the sides meet? 2. Is it going to infinity? 3. Is it oscillating?
For Advanced Learners (Extension):Constructing Graphs: Give students a set of conditions and have them draw a possible function.
Infinite Limits: Introduce the idea that if a graph goes to infinity from both sides, we sometimes say the limit is infinity even though infinity is not a real number (DNE).
AP College Board assessments