What is Limit of a sequence?

Example 11.2 (An $\varepsilon$–$N$ proof, written out once)

Claim: $u_n = \dfrac{n^2 + 1}{2n^2 - 3} \to \dfrac12$. First isolate the error:

Then dominate it by something simple: for $n \geq 2$, $2n^2 - 3 \geq n^2$, so the error is $\leq \frac{5}{2n^2} \leq \frac 5{2n}$. Given $\varepsilon > 0$, the Archimedean property provides $N \geq \max\bigl(2, \frac{5}{2\varepsilon}\bigr)$; for $n \geq N$ the error is $\leq \varepsilon$. Done. The closing insight: an $\varepsilon$–$N$ proof has exactly three moves — compute the error, bound it by a decreasing elementary expression, solve for the threshold — and after this chapter’s theorems (operations, squeeze) one almost never writes such a proof again: the theorems package the three moves once and for all.

Example 11.3 (Divergence to infinity, certified)

Claim: $u_n = n^2 - 100n \to +\infty$. Factor the dominant term: $u_n = n^2\bigl(1 - \frac{100}{n}\bigr) \geq \frac{n^2}{2}$ for $n \geq 200$. Given $M$, take $N = \max\bigl(200, \lceil\sqrt{2M}\rceil\bigr)$: for $n \geq N$, $u_n \geq \frac{n^2}{2} \geq M$. Two habits are on display: dominant-term factoring converts a competition ($n^2$ against $-100n$) into a single scale times a factor tending to $1$; and the threshold may be huge ($u_{100} = 0$, the sequence is even negative before $n = 100$) — divergence to $+\infty$ is a statement about the tail, indifferent to any finite amount of misbehavior.

Example 11.6 (Operations plus one algebraic trick)

Compute $\lim\,\bigl(\sqrt{n^2 + n} - n\bigr)$. The two pieces separately tend to $+\infty$: the operations theorem says nothing about their difference (an indeterminate form). Multiply by the conjugate:

Now everything converges: $\sqrt{1 + \frac1n} \to 1$, because $0 \leq \sqrt{1 + h} - 1 = \frac{h}{\sqrt{1+h} + 1} \leq h$ (conjugate again, then squeeze with $h = \frac1n$); then the operations theorem gives the limit $\frac{1}{1 + 1} = \frac12$. The closing insight: the operations theorem is not a calculator for all limits — indeterminate forms ($\infty - \infty$, $\frac00$, $0 \times \infty$, $1^\infty$) must first be transformed by algebra (conjugates, factoring the dominant term) until every piece converges; the systematic machine for resistant cases is the asymptotic expansion of Chapter 16.