Limites infiniti · Topos Uranos

Svmmarivm

Haec lectio fvnctiones inqvirit qvarvm valores sine margine crescvnt avt decrescvnt. Schema commvne definitionvm limitis locvlo qvi deerat complet, eo scilicet qvi valorem prope $\infty$ respicit, et limites infinitos, qvi modvs accvratvs svnt limitem non habendi, a ceteris divergentiae formis distingvit. Vnvm lemma, lemma inversi, limites infinitos ad limites nihilo aeqvales redvcit, ita vt regvlae svmmae, prodvcti et qvotientis ex regvlis algebrae limitvm dedvcantvr; sola comparatio, theorema de fvnctionibvs monotonis non circvmscriptis et compositio argvmentvm proprivm reqvirvnt. His dvo eventvs complentvr qvos lectiones svperiores imperfectos reliqverant: qvotiens cvivs denominator ad nihilvm tendit et fvnctiones rationales qvarvm nvmerator gradvm maiorem habet. Lectio etiam ostendit cvr formae indeterminatae regvlae non sint, et compositione cvm limitibvs infinitis finitvr.

Proposita discendi

Limites infinitos, in pvncto, ex vno latere et ad infinitvm, schemate provocationis et responsionis definire, eosqve a ceteris divergentiae formis distingvere.
Lemma inversi demonstrare et ex eo atqve ex comparatione regvlas svmmae, prodvcti et qvotientis cvm limitibvs infinitis dedvcere.
Formas indeterminatas agnoscere et exemplis ostendere eas qvemlibet exitvm dare posse.
Limites infinitos polynomiorvm, fvnctionvm rationalivm, fvnctionvm trigonometricarvm et compositionvm compvtare, et theorema de fvnctionibvs monotonis non circvmscriptis demonstrare.

Locvlvs qvi deerat

In lectione de limitibvs cvm variabilis ad infinitvm tendit schema commvne descriptvm est: omnis definitio limitis est provocatio de valoribvs cvm responsione de variabili, et solvm mvtatvr qvid sit prope vtrvmqve esse. Ad variabilem omnes formae iam adhibitae svnt: prope nvmervm $a$ , ex vno eivs latere, avt prope $\pm\infty$ . Ad valores vna tantvm: prope nvmervm $L$ . Haec lectio schema forma reliqva complet, in qva valor prope $\infty$ est, id est $f(x) > N$ , avt prope $-\infty$ , id est $f(x) < N$ .

DefinitioLimites infiniti

Sit $a$ pvnctvm accvmvlationis dominii $A$ fvnctionis $f$ . Dicitvr $f$ ad infinitvm tendere cvm $x$ ad $a$ tendit, et scribitvr $\lim_{x \to a} f(x) = \infty$ , si pro omni nvmero reali $N$ exsistit aliqvod $\delta > 0$ tale vt $f(x) > N$ pro omni $x$ ex $A$ cvm $0 < \abs{x - a} < \delta$ . Eodem modo, $\lim_{x \to a} f(x) = -\infty$ si pro omni $N$ exsistit aliqvod $\delta > 0$ tale vt $f(x) < N$ in illis $x$ .

Qvantificatoribvs scripta, definitio formam solitam habet; provocatio nvnc est semiplanvm svperivs, scilicet pvnctorvm qvorvm ordinata maior est qvam $N$ :

\lim_{x \to a} f(x) = \infty \leftrightarrow (\forall N)(\exists \delta > 0)(\forall x \in A)\left(0 < \abs{x - a} < \delta \rightarrow f(x) > N\right)

Variantes obtinentvr sola responsione mvtata, vt in lectionibvs svperioribvs: $\lim_{x \to a^+} f(x) = \infty$ si condicio valet pro $a < x < a + \delta$ ; $\lim_{x \to \infty} f(x) = \infty$ si exsistit aliqvod $M$ tale vt valeat pro $x > M$ ; et sic in ceteris casibvs. In iis qvae seqvvntvr, $P$ qvamlibet ex metis variabilis significat ( $a$ , $a^+$ , $a^-$ , $\infty$ avt $-\infty$ ), et «prope $P$ » significat in intervallo perforato, ex vno latere avt in semirecta, provt casvs fert. Omnes demonstrationes lectionis vnam tantvm proprietatem commvnem adhibent: dvae condiciones qvae prope $P$ valent simvl prope $P$ valent, si minor ex dvobvs radiis avt maior ex dvobvs marginibvs svmitvr. Theoremata de limitibvs finitis qvae in qvavis meta $P$ citantvr (vnicitas, circvmscriptio localis, conservatio signi, algebra limitvm et lemmata lectionis de algebra limitvm) in omnibvs metis valent per lectionem de limitibvs lateralibvs et per lemma inversionis lectionis svperioris, cvivs correspondentia $\delta = \frac{1}{M}$ inter marginem et radivm limites infinitos eodem modo transfert, qvia a forma provocationis non pendet.

Instandvm est in eo qvod scriptvra dicit. Si $\lim_{x \to a} f(x) = \infty$ , fvnctio limitem non habet in $a$ sensv lectionis de limite fvnctionvm vnivs variabilis realis, qvia prope $a$ circvmscripta non est, et circvmscriptio localis dicit fvnctionem limitem habentem circvmscriptam esse. Aeqvalitas cvm $\infty$ modvm accvratvm describit limitem non habendi. Dicitvr $f$ divergere in $P$ si ibi limitem finitvm non habet; ad $\infty$ vel ad $-\infty$ tendere dvo modi divergendi svnt, sed non soli. Fvnctio $\sin \frac{1}{x}$ in $0$ divergit et circvmscripta est; fvnctio $\frac{1}{x}\sin\frac{1}{x}$ in $0$ divergit, prope $0$ circvmscripta non est et tamen neqve ad $\infty$ neqve ad $-\infty$ tendit, qvia in pvnctis evanescit qvam proxime libet ad $0$ sitis.

PropositioInversa ipsivs

x^2

et ipsivs

x

in nihilo

$\lim_{x \to 0} \frac{1}{x^2} = \infty$ . Contra, $\lim_{x \to 0^+} \frac{1}{x} = \infty$ et $\lim_{x \to 0^-} \frac{1}{x} = -\infty$ , ita vt $\frac{1}{x}$ neqve ad $\infty$ neqve ad $-\infty$ tendat cvm $x$ ad $0$ tendit. Responsio ad provocationem $N$ est radix inversi ipsivs $N$ . In figvra, semiplanvm svperivs est provocatio; fascia verticalis ivxta $0$ , responsio.

Demonstratio

$\delta = \dato{d}{\frac{1}{\sqrt{N}}}$
Sit $N$ nvmervs realis; satis est $N > 0$ considerare, qvia qvi maiori provocationi respondet, omnibvs minoribvs respondet. Vt responsio svmitvr radix qvadrata eivs inversi.
$0 < x^2 < \frac{1}{N} \rightarrow \resaltar{\frac{1}{x^2} > N}$
Si $0 < \abs{x} < \delta$ , qvadratvm ipsivs $x$ positivvm est et minvs qvam $\frac{1}{N}$ ; inversis nvmerorvm positivorvm svmptis, inaeqvalitas invertitvr.
$0 < x < \frac{1}{N} \rightarrow \frac{1}{x} > N, \qquad -\frac{1}{N} < x < 0 \rightarrow \frac{1}{x} < -N$
Pro $\frac{1}{x}$ a dextra, propositio de inverso lectionis svperioris responsionem $\delta = \frac{1}{N}$ dat. A sinistra, $\frac{1}{x}$ est oppositvm ipsivs $\frac{1}{-x}$ , et $-x$ ad dextram ipsivs $0$ iacet: valores minores svnt qvam $-N$ .
$\lim_{x \to 0^+} \frac{1}{x} = \infty \neq -\infty = \lim_{x \to 0^-} \frac{1}{x}$
In omni intervallo perforato centri $0$ svnt valores ipsivs $\frac{1}{x}$ maiores qvam $1$ et minores qvam $-1$ : nvllvm $\delta$ provocationi $N = 1$ pro $\infty$ respondet, nec provocationi $N = -1$ pro $-\infty$ .