Limits of Trig Expressions

Teacher — unlock codes (cut off before distributing student cards)

Phase 1: MACLRN

Phase 2: DIVERG

Phase 3: FINITE

Phase 4: TAYLOR

∞ Activity Overview

Limits of Trig Expressions using Maclaurin Series · HL 5.13

Work at your group's vertical whiteboard. Phases unlock one at a time — show your teacher each completed phase to receive the next code. The main question carries 8 marks total.

⏱ ~25 min  ·  👥 Groups of 3  ·  HL only

Phase 1 · Warm-up

Tasks A & B

Task A — Familiar limit

Try small values of \(x\), then confirm with the standard result.

\[\lim_{x \to 0} \frac{\sin 5x}{x} = \;?\]

Task B — Harder denominator

What indeterminate form do you get? What technique can you use?

\[\lim_{x \to 0} \frac{\sin 3x}{x^3} = \;?\]

Does this limit exist?

Phase 1 · Warm-up

Task C — Bridge

Task C — When can two infinities cancel?

For what value of \(n\) does the limit below exist as a finite number? Show full reasoning — explain why only one value works.

\[\lim_{x \to 0} \left(\frac{\sin 3x}{x^3} + \frac{n}{x^2}\right)\]

Hint: expand \(\sin 3x\) as a Maclaurin series and divide by \(x^3\).

Code to unlock Phase 2:MACLRN

Phase 2 · 3 marks

Part (a) — Show that

The function

\[f(x) = m + \frac{\sin 3x}{x^3} + \frac{n}{x^2}, \quad m,n \in \mathbb{R}\]

Part (a) 3 marks

Show that a finite limit requires \(n = -3\).

Write a complete argument on the board. "Show" means your reasoning must be explicit — not just the answer.

Code to unlock Phase 3:DIVERG

Phase 3 · 5 marks

Part (b) — Find \(m\)

With \(n = -3\)

\[f(x) = m + \frac{\sin 3x}{x^3} - \frac{3}{x^2}\]

Part (b) 5 marks

Find \(m\) such that \(\lim_{x\to 0} f(x) = 5\).

Show every algebraic step. Exact form required.

Code to unlock Phase 4:FINITE

Phase 4 · Extension

Go Further

Extension 1

Replace \(\sin 3x\) with \(\sin 5x\).

What changes? Find the new \(n\) and the new \(m\) if the limit still equals 5.

Extension 2

L'Hôpital's route.

Solve part (b) using L'Hôpital's rule. How many applications are needed? Which method is more efficient?

Extension 3 — Generalise

Replace \(\sin 3x\) with \(\sin ax\).

Find \(n\) in terms of \(a\). Find \(\lim_{x\to 0} f(x)\) in terms of \(m\) and \(a\).

Code to complete:TAYLOR