• Quick retrieval from earlier learning (Rational functions and asymptotes; Find limits of rational functions as x → ∞). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. Find the horizontal asymptote of f(x) = (4x − 4)/(3x + 3). Give the y-value (fractions allowed). → 4/3 (1 1/3)Prompt: For large x, compare the leading terms of top and bottom.
2. Find the limit as x → ∞ of (4x² + 7x + 3)/(9x² + 4x − 1). Give a fraction if needed. → 4/9Prompt: For huge x, only the highest powers matter.
3. What happens to (3x² + 1)/(2x² − 3) as x → ∞? (0 / 3/2 / No finite limit (it grows without bound)) → 3/2 (1 1/2)Prompt: Compare the degree of the top with the degree of the bottom.

• Show a "think of a number" puzzle and ask the class to work out the starting number.
• Share the objective: Find vertical and horizontal asymptotes. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: f(x) = (4x − 9)/(x − 1). Find the vertical asymptote x = ?
• Step 1: x = 1
• Answer: 1
• Pause after each step and ask a student to say what comes next.

• Do these together. Students try each step on whiteboards before you reveal it.
• Use the hints as prompts rather than giving the answer.

1. f(x) = (x + 6)/(x − 2). Find the horizontal asymptote y = ? → 1Prompt: For large x, f(x) behaves like the ratio of the x terms.
2. What are the vertical asymptotes of f(x) = (x² + 1)/(4x² − 100)? (x = 5 and x = −5 / x = 5 / y = 1/4 / x = 25 and x = −25) → x = 5 and x = −5Prompt: Set the denominator equal to zero.

• 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.

• Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.

1. f(x) = (−5x − 8)/(x + 2). Find the vertical asymptote x = ? → -2Prompt: A vertical asymptote is where the denominator is zero (and the top isn't).
2. What are the vertical asymptotes of f(x) = (5x² + 1)/(3x² − 108)? (x = 6 and x = −6 / y = 5/3 / There are none / x = 36 and x = −36) → x = 6 and x = −6Prompt: Set the denominator equal to zero.

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.