Activity 01
Pair Graphing: Limit Visualisation
Pairs use graphing calculators to plot polynomial and rational functions, zooming near critical points. They record limit values from tables and graphs, then compare with algebraic results. Discuss discrepancies to confirm understanding of asymptotes.
Analyze strategies for evaluating limits of rational functions that result in indeterminate forms.
Facilitation TipDuring Pair Graphing, ask students to trace the curve with their finger as they approach the point from both sides to notice where left and right limits meet.
What to look forPresent students with three rational functions. Ask them to: 1. Identify which function, if any, results in an indeterminate form when the limit is evaluated by direct substitution. 2. For that function, show the steps to factorise and find the limit. 3. For another function, explain whether the limit exists, does not exist, or is infinite.
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Activity 02
Small Group Puzzle: Indeterminate Forms
Distribute cards with rational functions showing 0/0 forms. Groups simplify step-by-step on mini-whiteboards, predict limits, and verify with substitution. Rotate roles for factorisation and checking.
Differentiate between limits that exist, do not exist, and are infinite.
Facilitation TipFor the Small Group Puzzle, give each group only one sheet with a 0/0 form so they must collaborate to factorise and reach the simplified limit.
What to look forOn a small slip of paper, ask students to write down: 1. One strategy for evaluating a limit that results in 0/0. 2. The equation of a vertical asymptote for a given rational function, and justify their answer using limits.
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Activity 03
Whole Class Debate: Limit Existence
Project functions with oscillating or one-sided limits. Class votes on existence, then debates evidence from graphs and tables. Teacher facilitates consensus on criteria for DNE limits.
Predict the behavior of a rational function near a vertical asymptote using limits.
Facilitation TipIn the Whole Class Debate, write key phrases like ‘limit exists’, ‘limit is infinite’, and ‘limit does not exist’ on the board so students can categorise their findings as they speak.
What to look forPose the question: 'When evaluating the limit of a rational function f(x)/g(x) at x=a, what does it mean if f(a)=0 and g(a)=0? How does this differ from the case where f(a) is non-zero and g(a)=0?' Facilitate a class discussion comparing indeterminate forms with infinite limits.
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Activity 04
Individual Challenge: Asymptote Prediction
Students receive rational functions, sketch graphs mentally, and predict vertical asymptotes via limits. They test predictions on graph paper, self-assess against answer keys.
Analyze strategies for evaluating limits of rational functions that result in indeterminate forms.
Facilitation TipFor Individual Challenge, provide graph paper with axes already labelled and ask students to draw the asymptotes after computing the limit.
What to look forPresent students with three rational functions. Ask them to: 1. Identify which function, if any, results in an indeterminate form when the limit is evaluated by direct substitution. 2. For that function, show the steps to factorise and find the limit. 3. For another function, explain whether the limit exists, does not exist, or is infinite.
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Generate Complete Lesson→A few notes on teaching this unit
Teachers find that starting with polynomials builds confidence because direct substitution works without surprises. When moving to rational functions, it is better to introduce indeterminate forms with simple examples first, then progress to higher-degree polynomials. Avoid rushing to rules; instead, let students discover why cancellation works by seeing the hole in the graph. Research shows that students who draw graphs while working with limits retain concepts longer than those who only compute symbolically.
By the end, students should confidently evaluate limits for polynomials, simplify rational functions to resolve indeterminate forms, and distinguish between limits that exist, do not exist, or tend to infinity. They will also explain their reasoning using graphs and algebraic steps.
Watch Out for These Misconceptions
During Pair Graphing, watch for students who assume the limit equals the function value even when the graph shows a hole.
Ask partners to mark the hole on their graph and write the limit value separately, then discuss why the function value and limit can differ in the same exercise.
During Small Group Puzzle, watch for groups who conclude that a 0/0 form always means the limit is zero without simplifying.
Have groups present their simplified expression and compare it with the original function, highlighting that the hole’s y-coordinate may not be zero.
During Whole Class Debate, watch for students who confuse infinite limits with non-existent limits.
Use the shared graph to trace the curve as it climbs toward infinity and write the notation lim(x→a) f(x) = ∞ clearly on the board to separate this from cases where left and right limits differ.
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