Math lesson plan · 12th grade (Year 13 in England) · 30 minutes
Find limits of rational functions as x → ∞
Calculus
CCSS HSF.IF.C.7.DEngland: A-level Pure Mathematics
Objective and success criteria
Students will: Find limits of rational functions as x → ∞.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Rational functions and asymptotes
Leads to: Find vertical and horizontal asymptotes
Materials: Graphing tool or graph paper, Scientific calculators, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 4 min
- Quick retrieval from earlier learning (Rational functions and asymptotes). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find the vertical asymptote of f(x) = (x − 3)/(x − 1). Give the x-value. → 1Prompt: The function has no value where the denominator is zero.
- Find the horizontal asymptote of f(x) = (6x − 1)/(2x − 4). Give the y-value (fractions allowed). → 3Prompt: For large x, compare the leading terms of top and bottom.
- Find the vertical asymptote of f(x) = (x − 5)/(x + 2). Give the x-value. → -2Prompt: The function has no value where the denominator is zero.
Hook · 3 min
- Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
- Share the objective: Find limits of rational functions as x → ∞. Read the success criteria together.
I do: model it · 7 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Find the limit as x → ∞ of (x² + 2x + 3)/(2x² + 7x − 1). Give a fraction if needed.
- Step 1: 1x²/2x² → 1/2
- Answer: 1/2
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 6 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Find the limit as x → ∞ of (9x² + 8x + 3)/(8x² − 3x − 1). Give a fraction if needed. → 9/8 (1 1/8)Prompt: For huge x, only the highest powers matter.
- What happens to (4x³ + 2)/(4x² − 3) as x → ∞? (0 / 1 / No finite limit (it grows without bound)) → No finite limit (it grows without bound)Prompt: Compare the degree of the top with the degree of the bottom.
You do: independent practice · 7 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- What happens to (2x² + 1)/(6x² − 3) as x → ∞? (0 / 1/3 / No finite limit (it grows without bound)) → 1/3Prompt: Compare the degree of the top with the degree of the bottom.
- Find the limit as x → ∞ of (x² + 9x + 3)/(4x² − 6x − 1). Give a fraction if needed. → 1/4Prompt: For huge x, only the highest powers matter.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Rational functions and asymptotes.
- Use concrete materials: graphing tool or graph paper.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Move on to: Find vertical and horizontal asymptotes.
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.
Free lesson plan from eurekami.com/lesson-plan-builder