Limits¶
Recall (doc/postdoc). Not a first course.
Job: topological limit in \(\mathbb{R}\); \(\varepsilon\)–\(\delta\) as metric continuity of nets/filters — used, not taught.
Core¶
\[
\lim_{x\to a}f(x)=L\iff \forall\varepsilon>0\ \exists\delta>0:\ 0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon
\]
| Variant | Criterion |
|---|---|
| One-sided | restrict \(x<a\) or \(x>a\) |
| Two-sided exists | left \(=\) right |
| At \(\infty\) | $\forall\varepsilon\ \exists M:\ x>M\Rightarrow |
| Continuity at \(a\) | \(\lim_{x\to a}f=f(a)\) |
Algebra: limits respect \(+,\cdot,/\) (denom limit \(\neq0\)) when exist.
Indeterminate forms: \(0/0,\infty/\infty,0\cdot\infty,\infty-\infty,1^\infty,0^0,\infty^0\) — rewrite or L'Hôpital (when \(f,g\) diffble, \(g'\neq0\), limit of \(f'/g'\) exists).
Standard: \(\lim_{x\to0}\frac{\sin x}{x}=1\); polynomials continuous; rational continuous on domain.
Used later for¶
- Derivatives: difference quotients
- Real analysis: sequential char., Heine–Cantor
- Topology: general continuity
- Complex analysis: complex limits path-independent for holomorphy
Check¶
Does L'Hôpital apply to \(\lim_{x\to0}\frac{x^2\sin(1/x)}{x}\)?
Form \(0/0\), but \(g'=1\), \(f'/g'\) limit need not exist in naive setup for oscillating \(f\); better: squeeze \(|f/g|\le|x|\to0\). Hypotheses on existence of \(\lim f'/g'\) are essential.