Stage review: limits from the definition

This review asks for two proofs from the definitions: an ε–N proof for a sequence, and an ε–δ proof for a function. As always, each claim is tested first.

Take $a_n = \frac{5n - 2}{2n + 1}$. Its leading terms are $5n$ and $2n$, so the rule of lesson 2.1.3 points to the limit $\frac{5}{2}$.

The terms fall short of $\frac{5}{2}$ by $\frac{3}{2}$ at $n = 1$, $\frac{3}{14}$ at $n = 10$ and $\frac{3}{134}$ at $n = 100$. For the tolerances 0.1, 0.01 and 0.001, the last term outside the band is $n = 22$, 224 and 2249.

A wrong limit fails. Measured from 2.49, the distance settles near 0.01, not 0: 0.009978 at $n = 10^5$. So with $\varepsilon = 0.005$, the terms never stay within $\varepsilon$ of 2.49.

Now a function: $f(x) = \frac{1}{x}$ near $x = 2$, where substituting (lesson 2.1.1) gives $\frac{1}{2}$. The danger is $x = 0$, where $\frac{1}{x}$ blows up (lesson 2.1.2).

The cell fixes $\varepsilon = 0.1$ and tries five windows around 2. For each it reports the worst distance from $\frac{1}{2}$ among 1,998 inputs, and whether that is below $\varepsilon$.

The windows 0.3, 0.2 and 0.1 work, and the window 0.5 fails, with a worst distance of 0.166445.

The window 2 reaches almost to 0, and there the worst distance is 499.5. A proof must keep the window away from 0.