One-Sided Limits Comparison
About This MicroSim
This visualization helps students understand the relationship between one-sided limits and two-sided limits. A two-sided limit exists if and only if both one-sided limits exist and are equal.
Function Presets
- Jump at x=1: Left and right limits differ → two-sided limit DNE
- Continuous at x=2: Left and right limits equal → two-sided limit exists
- Hole at x=2: Left and right limits equal → two-sided limit exists (even with a hole)
- Jump at x=0: Left and right limits differ → two-sided limit DNE
Key Concept
\[\lim_{x \to c} f(x) = L \text{ exists } \iff \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L\]
How to Use
- Function Selector: Choose from different piecewise functions
- Distance Slider: Control how close both points approach the target
- Animate: Watch both approaches simultaneously
- Info Panel: See the numerical values of both limits and whether the two-sided limit exists
Lesson Plan
Learning Objectives
After using this MicroSim, students will be able to:
- Compare left-hand and right-hand limits graphically
- Determine whether a two-sided limit exists based on one-sided limits
- Explain why jump discontinuities prevent limits from existing
Suggested Activities
- Sort the Functions: Categorize each preset as "limit exists" or "limit DNE"
- Predict Before Selecting: For each function name, predict whether the limit will exist
- Create Your Own: Describe a function where the left limit is 3 and right limit is 5
Assessment Questions
- If \(\lim_{x \to 5^-} f(x) = 7\) and \(\lim_{x \to 5^+} f(x) = 7\), what is \(\lim_{x \to 5} f(x)\)?
- If \(\lim_{x \to 2^-} f(x) = 4\) and \(\lim_{x \to 2^+} f(x) = 6\), does \(\lim_{x \to 2} f(x)\) exist?
- Can a function have a two-sided limit at a point where it has a hole?
Embedding
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