Level 5: Limits

Infinity and strange math · level 5 of 8 · rough age 12-13 (a guide for parents; anyone can start anywhere)

In level 4 you added up endless sums and drew shapes that never stop growing. This level asks what "endless" really means, and answers with the idea of a limit: the number that a never-ending list closes in on. With it you can settle Zeno's race to the exact second, prove that \(0.333\ldots\) is exactly \(\frac13\), see why \(\infty\) is not a number and why we never divide by zero, and spot the traps in some famous paradoxes.

Lesson	You will be able to
Sequences that settle down	Explain what a limit is with the closeness game, find limits of simple sequences, and show that some sequences have none
Zeno answered	Add up Zeno's infinitely many stages to find exactly when Achilles passes the tortoise, and say where Zeno's argument goes wrong
Infinity is not a number	Show why \(\infty - \infty\) and \(\frac{\infty}{\infty}\) have no single answer, and why \(6 \div 0\) has no answer and \(0 \div 0\) has every answer
Approaching asymptotes	Find the horizontal and vertical asymptotes of curves like \(y = \frac1x\), and show that a curve can cross its asymptote
Repeating decimals and limits	Explain why an endless decimal means a limit, prove \(0.999\ldots = 1\), and turn any repeating decimal into a fraction
Paradoxes of the infinite	Analyse Thomson's lamp, the Ross-Littlewood urn and the staircase, and say exactly why each answer depends on the setup

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