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Vertical Asymptote Explorer

Run the Vertical Asymptote Explorer Fullscreen

Edit the Vertical Asymptote Explorer with the p5.js editor

Iframe Example

Place the following line in your website to include this MicroSim:

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<iframe src="https://dmccreary.github.io/calculus/sims/vertical-asymptote/main.html" height="482px" scrolling="no"></iframe>

Description

This MicroSim helps students understand vertical asymptotes by visualizing how function values behave as x approaches the asymptote from both sides. Key features include:

  • Multiple function examples: Choose from four rational functions including 1/x, 1/x^2, and functions with multiple asymptotes
  • Animated approach: Watch points trace along the curve as they get closer to the asymptote
  • Direction indicators: Arrows show whether the function heads toward positive or negative infinity
  • One-sided limit notation: The info panel displays the formal one-sided limit notation for each direction
  • Adjustable approach distance: Use the slider to manually control how close to the asymptote the approach points get

The color coding helps distinguish the two approaches: - Teal: Approaching from the left (x approaching a from below) - Orange: Approaching from the right (x approaching a from above)

Delta Moment

"See how I get closer and closer to that vertical line but can NEVER cross it? Watch my y-value--it's heading off to infinity! The slope under my wheels is getting steeper... and steeper... WHEEE!"

Lesson Plan

Learning Objective

Students will explain how one-sided limits determine the behavior of a function near its vertical asymptotes.

Bloom's Taxonomy Level

Understand (L2) - Students interpret visual representations to explain mathematical behavior.

Prerequisites

  • Understanding of function notation
  • Basic knowledge of limits
  • Familiarity with rational functions

Warm-Up (5 minutes)

  1. Start with f(x) = 1/x selected
  2. Ask: "What happens to the function value when x is very small and positive? Very small and negative?"
  3. Let students predict before using the animation

Guided Exploration (10 minutes)

  1. Same-sign behavior (1/x^2):
  2. Switch to f(x) = 1/x^2
  3. Animate the approach
  4. Ask: "Do both sides go to the same infinity or different infinities?"

  5. Discuss why (squaring makes everything positive)

  6. Different-sign behavior (1/x):

  7. Switch back to f(x) = 1/x
  8. Animate and compare

  9. Ask: "Why do the two sides go in opposite directions?"

  10. Multiple asymptotes:

  11. Select f(x) = x/((x-2)(x+1))
  12. Explore behavior at both x = -1 and x = 2
  13. Note: The function selector shows the first asymptote by default

Independent Practice (10 minutes)

Have students complete the following:

  1. For each function, predict the one-sided limit behavior before animating
  2. Sketch the function near its asymptote(s)
  3. Write the formal one-sided limit notation

Assessment Questions

  1. If the left-hand limit is +infinity and the right-hand limit is -infinity, can the two-sided limit exist? Why or why not?

  2. For f(x) = 1/x^2, explain why both one-sided limits equal +infinity even though the function approaches from different directions.

  3. Given a new rational function, how would you predict where the vertical asymptotes occur?

Extension Activities

  • Challenge: Can you create a function where the left limit is +infinity and the right limit is a finite number? (Discuss piecewise functions)
  • Connection to continuity: How do vertical asymptotes relate to discontinuities?

References