haya.Learning made clear
Limits of functions and sequences sheet
Structured-answer frame

Evaluate (a) lim (x → 3) (x² − 9)/(x − 3); (b) lim (x → ∞) (4x² + 3x)/(2x² − 5); (c) explain what happens to AC = 20 + 5,000/x as x → ∞.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · (a) Substitute first. Gives 0/0, so the expression must be simplified.
  2. 2 · Factorise and cancel. (x + 3)(x − 3)/(x − 3) = x + 3, so the limit is 6.
  3. 3 · (b) Divide by the highest power, x². (4 + 3/x)/(2 − 5/x²) → 4/2 = 2.
  4. 4 · (c) Split the average cost. AC = 20 + 5,000/x; as x → ∞, 5,000/x → 0, so AC → 20.
  5. 5 · Business link. Fixed cost per unit vanishes at high output; unit cost approaches the Rs. 20 variable cost.