Limits from Graphs Practice
About This MicroSim
Practice the essential skill of reading limits directly from graphs. This skill is critical for understanding limits conceptually and appears frequently on AP Calculus exams.
What You'll Practice
- Left-hand limits: \(\lim_{x \to c^-} f(x)\)
- Right-hand limits: \(\lim_{x \to c^+} f(x)\)
- Two-sided limits: \(\lim_{x \to c} f(x)\) (or DNE if one-sided limits differ)
Graph Types by Difficulty
- Basic: Continuous functions (limit = function value)
- Intermediate: Holes (removable discontinuities)
- Challenge: Jump discontinuities and asymptotes
How to Use
- Select Difficulty: Choose Basic, Intermediate, or Challenge
- Analyze the Graph: Look at behavior as x approaches the target
- Enter Limits: Click each input box and type the limit value
- Check Your Work: Get feedback on your answers
- Trace Points: Hover over the graph to see exact coordinates
Input Tips
- Click on an input box to select it
- Use Tab to move between inputs
- Type "DNE" for two-sided limits that don't exist
- Press Enter to check your answer
Lesson Plan
Learning Objectives
After using this MicroSim, students will be able to:
- Determine limit values from graphical representations
- Distinguish between function values and limit values
- Identify when two-sided limits exist vs. when they don't exist
Key Skills
When reading limits from graphs: - Look at where the curve is heading, not where a point might be - Check both sides for two-sided limits - Open circles indicate holes (the limit may still exist!) - Jump discontinuities mean the two-sided limit doesn't exist
Suggested Activities
- Start with Basic: Master continuous functions before moving to discontinuities
- Verbalize Your Process: Say "as x approaches 2 from the left, y is heading toward..."
- Challenge Mode: Can you get 5 challenge problems correct in a row?
Assessment Questions
- If you see an open circle at (3, 5) and a solid dot at (3, 2), what is \(\lim_{x \to 3} f(x)\)?
- How can you tell from a graph that a two-sided limit doesn't exist?
- What's the difference between \(f(c)\) and \(\lim_{x \to c} f(x)\)?
Embedding
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