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Math lesson plan · 12th grade (Year 13 in England) · 60 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 · 6 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.
  1. Find the vertical asymptote of f(x) = (x + 5)/(3x + 12). Give the x-value. → -4Prompt: The function has no value where the denominator is zero.
  2. Find the horizontal asymptote of f(x) = (2x)/(x − 1). Give the y-value (fractions allowed). → 2Prompt: For large x, compare the leading terms of top and bottom.
  3. Find the horizontal asymptote of f(x) = (3x − 1)/(2x − 2). Give the y-value (fractions allowed). → 3/2 (1 1/2)Prompt: For large x, compare the leading terms of top and bottom.

Hook · 6 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 · 12 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 · 12 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. Find the limit as x → ∞ of (5x² + 6x + 3)/(8x² + x − 1). Give a fraction if needed. → 5/8Prompt: For huge x, only the highest powers matter.
  2. What happens to (6x + 1)/(6x² − 3) as x → ∞? (0 / 1 / No finite limit (it grows without bound)) → 0Prompt: Compare the degree of the top with the degree of the bottom.

You do: independent practice · 17 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 8 questions are on the student sheet below, with answers in the key.

Exit ticket · 7 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. Find the limit as x → ∞ of (3x² + 6x + 3)/(8x² + 8x − 1). Give a fraction if needed. → 3/8Prompt: For huge x, only the highest powers matter.
  2. What happens to (2x + 1)/(3x² − 3) as x → ∞? (0 / 2/3 / No finite limit (it grows without bound)) → 0Prompt: Compare the degree of the top with the degree of the bottom.

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

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Find limits of rational functions as x → ∞

Name

Date

Score/ 8

  1. Question 1

    Find the limit as x → ∞ of (−5x² + 9x + 3)/(4x² + 4x − 1). Give a fraction if needed.

    Answer(as a fraction)

  2. Question 2

    What happens to (6x² + 1)/(3x² − 3) as x → ∞?

    • A. 0
    • B. 2
    • C. No finite limit (it grows without bound)

  3. Question 3

    Find the limit as x → ∞ of (6x² + 8x + 3)/(4x² + 8x − 1). Give a fraction if needed.

    Answer(as a fraction)

  4. Question 4

    Find the limit as x → ∞ of (3x² + 4x + 3)/(2x² − x − 1). Give a fraction if needed.

    Answer(as a fraction)

  5. Question 5

    What happens to (5x³ + 2)/(4x² − 3) as x → ∞?

    • A. 0
    • B. 5/4
    • C. No finite limit (it grows without bound)

  6. Question 6

    Find the limit as x → ∞ of (−7x² + 2x + 3)/(9x² + 4x − 1). Give a fraction if needed.

    Answer(as a fraction)

  7. Question 7

    What happens to (3x + 1)/(x² − 3) as x → ∞?

    • A. 0
    • B. 3
    • C. No finite limit (it grows without bound)

  8. Question 8

    Find the limit as x → ∞ of (4x² − 9x + 3)/(5x² + 7x − 1). Give a fraction if needed.

    Answer(as a fraction)

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Answer key: Find limits of rational functions as x → ∞

  1. 1. -5/4

  2. 2. B. 2

  3. 3. 3/2

  4. 4. 3/2

  5. 5. C. No finite limit (it grows without bound)

  6. 6. -7/9

  7. 7. A. 0

  8. 8. 4/5