BTC Activity · IBDP Mathematics AA · HL · 5.13

Limits of Trig Expressions
using Maclaurin Series

Topic 5 · Calculus · Groups of 3 · Vertical Non-Permanent Surfaces · ~25 min
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Warm-up Tasks

5.13 · Standard limits · Maclaurin series intro
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Vertical Non-Permanent SurfaceWork these tasks standing at your group's whiteboard. One marker, take turns writing.
Task A
A familiar limit

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

\[\lim_{x \to 0} \frac{\sin 5x}{x}\]
What does this equal? Why?
Task B
A harder denominator
\[\lim_{x \to 0} \frac{\sin 3x}{x^3}\]
What happens when you substitute directly? What indeterminate form do you get?
Task C — bridge task
When can two infinities cancel out?
\[g(x) = \frac{\sin 3x}{x^3} + \frac{n}{x^2}\]
For what value of \(n\) does \(\displaystyle\lim_{x \to 0} g(x)\) exist as a finite number?

Show your full reasoning on the board. Be ready to explain why only one value of \(n\) works.

🔐 Enter teacher code to unlock Phase 2
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The IB Question — Part (a)

5.13 · Show that \(n = -3\) is necessary · 3 marks
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Complete Phase 1 to unlock

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The IB Question — Part (b)

5.13 · Find \(m\) · 5 marks
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Complete Phase 2 to unlock

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Extension — Go Further

5.13 · Generalisation · L'Hôpital's rule
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Complete Phase 3 to unlock

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Activity Complete!

You've worked through all four phases.
Get ready for the gallery walk and class consolidation.

After consolidation, write your own notes from memory — not from the board.