Unit 02 · Lesson 08 · Week 6 · Limits: Numerical, Graphical & Algebraic Reasoning
Limits at Infinity and End Behavior
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 limits at infinity and end behavior through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in limits at infinity and end behavior.
- 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 can values near an input reveal behavior at that input—even when direct substitution fails?
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
Limits at Infinity and End Behavior
Teacher Answer Key · 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.
Answer: x=3
Reasoning: The denominator is zero at that input and does not cancel.
Reasoning: The denominator is zero at that input and does not cancel.
For f(x)=(3x²+1)/(x²−4), find the horizontal asymptote.
Answer: y=3
Reasoning: Equal degrees give the ratio of leading coefficients.
Reasoning: Equal degrees give the ratio of leading coefficients.
As x→∞, evaluate 5/x.
Answer: 0
Reasoning: The denominator grows without bound.
Reasoning: The denominator grows without bound.
Can a function cross a horizontal asymptote?
Answer: Yes
Reasoning: A horizontal asymptote describes end behavior.
Reasoning: A horizontal asymptote describes end behavior.
State the right-hand limit of 1/x as x→0⁺.
Answer: +∞
Reasoning: Positive small denominators produce large positive values.
Reasoning: Positive small denominators produce large positive values.
Distinguish a vertical from a horizontal asymptote.
Answer: Vertical describes behavior near a finite input; horizontal describes end behavior.
Reasoning: The directions of approach differ.
Reasoning: The directions of approach differ.