Unit 03 · Lesson 07 · Week 9 · Continuity & Asymptotic Behavior
Continuity, Asymptotes & Model Reliability
Infinite limits and limits at infinity distinguish local unbounded behavior from long-run behavior. Vertical asymptotes arise near finite inputs; horizontal or slant behavior describes what happens as the input magnitude grows. This lesson emphasizes continuity, asymptotes & model reliability through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in continuity, asymptotes & model reliability.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do continuity conditions connect local behavior, global guarantees, and model reliability?
Visual model
Infinite limits and limits at infinity distinguish local unbounded behavior from long-run behavior. Vertical asymptotes arise near finite inputs; horizontal or slant behavior describes what happens as the input magnitude grows.
\lim_{x\to a}f(x)=\pm\infty\lim_{x\to\infty}f(x)=Lx=a
Worked reasoning
Misconception check: A graph can cross a horizontal asymptote. The asymptote describes end behavior, not a forbidden output.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Continuity, Asymptotes & Model Reliability
Student Worksheet · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
For f(x)=1/(x−3), identify the vertical asymptote.
For f(x)=(3x²+1)/(x²−4), find the horizontal asymptote.
As x→∞, evaluate 5/x.
Can a function cross a horizontal asymptote?
State the right-hand limit of 1/x as x→0⁺.
Distinguish a vertical from a horizontal asymptote.
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.