Infinitesima et infinita aeqvivalentia

Svmmarivm

Haec lectio partem de limitibvs arte compvtandi clavdit qvae fere omnia svperiora conivngit: comparatione fvnctionvm qvae ad nihilvm vel ad infinitvm tendvnt. Definivntvr $o$ parvvm, $O$ magnvm, idem ordo et aeqvivalentia, et demonstratvr in prodvcto vel qvotiente qvemqve factorem alio aeqvivalenti svbstitvi posse limite non mvtato. Limites trigonometrici, exponentialis, logarithmi et potentiarvm, et polynomiorvm, in lectionibvs svperioribvs demonstrati, in tabvlam aeqvivalentivm colligvntvr, qva limites vno obtvtv compvtari possvnt qvi antea plvres transformationes postvlabant. Lectio etiam ostendit errorem hvivs artis freqventissimvm, svbstitvtionem intra svmmam, qvando svmma tamen svbstitvi possit, et qvomodo potentiae prope nihilvm et procvl ab origine comparentvr.

Proposita discendi

$o$ parvvm, $O$ magnvm, evndem ordinem et aeqvivalentiam fvnctionvm in pvncto, ex vno latere vel ad infinitvm definire, et earvm proprietates demonstrare.
Theorema svbstitvtionis aeqvivalentivm in prodvctis et qvotientibvs demonstrare, et agnoscere cvr in svmmis non valeat.
Tabvlam aeqvivalentivm in nihilo demonstrare eaqve cvm compositione vti ad limites compvtandos et ad ordinem et partem principalem infinitesimi determinandos.
Infinita et infinitesima potentiarvm comparare, et aeqvivalentiam ab exsistentia asymptoti distingvere.

Fvnctiones comparare

Lectiones de theoremate compressionis et de limitibvs infinitis ostendervnt formas $\frac{0}{0}$ et $\frac{\infty}{\infty}$ exitvm certvm non habere: limes qvotientis dvarvm fvnctionvm qvae ad nihilvm tendvnt pendet ex celeritate qva eo tendvnt. Consilivm hvivs lectionis est eam celeritatem metiri, qvamqve fvnctionem cvm alia simpliciore comparando. Vt in lectione de limitibvs infinitis, $P$ metam variabilis significat ( $a$ , $a^+$ , $a^-$ , $\infty$ avt $-\infty$ ), et «prope $P$ » significat in intervallo perforato, ex vno latere avt in semirecta. Si fvnctio solvm in copia $A$ definita est, cvivs $a$ pvnctvm accvmvlationis est, «prope $a$ » significat in pvnctis ipsivs $A$ intervalli perforati centri $a$ , et limites svnt ii lectionis de limite fvnctionvm vnivs variabilis realis, in $A$ svmpti; definitiones qvae seqvvntvr eodem modo in hoc casv adhibentvr.

DefinitioComparatio fvnctionvm

Sint $f$ et $g$ fvnctiones prope $P$ definitae, cvm $g(x) \neq 0$ prope $P$ .

$f$ neglegibilis est prae $g$ in $P$ , et scribitvr $f = \operatorname{o}(g)$ , qvod legitvr « $f$ est o parvvm ipsivs $g$ », si $\lim_{x \to P} \frac{f(x)}{g(x)} = 0$ .
$f$ per $g$ circvmscripta est in $P$ , et scribitvr $f = \operatorname{O}(g)$ , qvod legitvr « $f$ est O magnvm ipsivs $g$ », si exsistit $C > 0$ tale vt $\abs{f(x)} \leq C\abs{g(x)}$ prope $P$ ; id est, si qvotiens $\frac{f(x)}{g(x)}$ prope $P$ circvmscriptvs est.
$f$ et $g$ eivsdem ordinis svnt in $P$ si $\frac{f(x)}{g(x)}$ ad nvmervm a nihilo diversvm tendit.
$f$ et $g$ aeqvivalentes svnt in $P$ , et scribitvr $f \sim g$ , si $\lim_{x \to P} \frac{f(x)}{g(x)} = 1$ .

Fvnctio qvae ad $0$ in $P$ tendit infinitesimvm in $P$ appellatvr, et qvae ad $\infty$ vel ad $-\infty$ tendit, infinitvm. Infinitesimvm in $0$ ordinem $k > 0$ habet si eivsdem ordinis est ac $x^k$ (in $0^+$ , si $x^k$ solvm pro $x > 0$ definita est), et tvm, si $\frac{f(x)}{x^k}$ ad $c$ tendit, fvnctio $c\,x^k$ eivs pars principalis est: $f \sim c\,x^k$ .

Vocabvlvm «infinitesimvm» non nvmervm minimvm significat, sed fvnctionem qvae ad nihilvm tendit; scriptvra $\operatorname{o}(g)$ non fvnctionem certam significat, sed qvamlibet earvm qvae prae $g$ neglegibiles svnt. Exempli gratia, $x^2 = \operatorname{o}(x)$ in $0$ , qvia $\frac{x^2}{x} = x$ ad $0$ tendit; contra, $x = \operatorname{o}(x^2)$ in $\infty$ , qvia $\frac{x}{x^2} = \frac{1}{x}$ ad $0$ tendit. Idem par fvnctionvm pro meta contrario modo comparatvr. Idem ordo et aeqvivalentia re vera vna comparatio svnt: si $f$ et $g$ eivsdem ordinis svnt qvotiente ad $c$ tendente, tvm $f \sim c\,g$ .

PropositioProprietates comparationis

In eadem meta $P$ :

aeqvivalentia reflexiva, symmetrica et transitiva est: $f \sim f$ ; si $f \sim g$ , tvm $g \sim f$ ; et si $f \sim g$ et $g \sim h$ , tvm $f \sim h$ ;
$f \sim g$ si et solvm si $f - g = \operatorname{o}(g)$ ; praesertim, si $h = \operatorname{o}(g)$ , tvm $g + h \sim g$ ;
si $f = \operatorname{o}(h)$ et $g = \operatorname{o}(h)$ , tvm $f + g = \operatorname{o}(h)$ et $c\,f = \operatorname{o}(h)$ pro omni reali $c$ .

Qvaeqve affirmatio est regvla ex algebra limitvm ad qvotientem adhibita.

Demonstratio

$\frac{g}{f} = \frac{1}{\frac{f}{g}} \to 1, \qquad \frac{f}{h} = \frac{f}{g} \cdot \frac{g}{h} \to 1 \cdot 1 = \resaltar{1}$
Qvotiens ipsivs $f$ per se ipsam $1$ valet. Si $\frac{f}{g}$ ad $1$ tendit, eivs inversvm ad $\frac{1}{1} = 1$ tendit, per regvlam inversi. Si $\frac{f}{g}$ et $\frac{g}{h}$ ad $1$ tendvnt, etiam eorvm prodvctvm, per regvlam prodvcti.
$\frac{f - g}{g} = \frac{f}{g} - 1 \to 0 \Leftrightarrow \frac{f}{g} \to \resaltar{1}$
Qvotiens ipsivs $f - g$ per $g$ est $\frac{f}{g} - 1$ , qvi ad $0$ tendit si et solvm si $\frac{f}{g}$ ad $1$ tendit. Cvm $f = g + h$ , differentia est $h$ .
$\frac{f + g}{h} = \frac{f}{h} + \frac{g}{h} \to 0 + 0 = \resaltar{0}$
Si $\frac{f}{h}$ et $\frac{g}{h}$ ad $0$ tendvnt, etiam eorvm svmma et prodvctvm vtrivsqve per constantem, per regvlam combinationis linearis.

$O$ magnvm non postvlat vt qvotiens limitem habeat, sed solvm vt non sine modo crescat. Exempli gratia, $\sin x = \operatorname{O}(1)$ in $\infty$ , qvamvis $\sin x$ ibi limitem non habeat, et $x \sin \frac{1}{x} = \operatorname{O}(x)$ in $0$ , qvia $\abs{x \sin \frac{1}{x}} \leq \abs{x}$ . Ad limites compvtandos svfficit $o$ parvvm; $O$ magnvm ad errores approximationis metiendos serviet, vt evm polynomii Tayloriani, qvi in Calcvlo II tractabitvr. Eivs regvlae hae svnt.

PropositioRegvlae O magni

In eadem meta $P$ :

si $\frac{f}{g}$ limitem finitvm in $P$ habet, tvm $f = \operatorname{O}(g)$ ; praesertim, si $f = \operatorname{o}(g)$ vel si $f \sim g$ ;
si $f = \operatorname{O}(h)$ et $g = \operatorname{O}(h)$ , tvm $f + g = \operatorname{O}(h)$ et $c\,f = \operatorname{O}(h)$ pro omni reali $c$ ;
si $f_1 = \operatorname{O}(g_1)$ et $f_2 = \operatorname{O}(g_2)$ , tvm $f_1 f_2 = \operatorname{O}(g_1 g_2)$ ;
si $f_1 = \operatorname{O}(g_1)$ et $f_2 = \operatorname{o}(g_2)$ , tvm $f_1 f_2 = \operatorname{o}(g_1 g_2)$ .

Qvaeqve regvla inaeqvalitas inter valores absolvtos est; vltima marginem cvm qvotiente ad nihilvm tendente conivngit.

Demonstratio

$\abs{\frac{f(x)}{g(x)}} \leq \abs{\frac{f(x)}{g(x)} - L} + \abs{L} < \resaltar{\abs{L} + 1}$
Si $\frac{f}{g}$ ad $L$ tendit, provocatio $\varepsilon = 1$ responsionem habet: prope $P$ qvotiens a $L$ minvs qvam $1$ distat, et per inaeqvalitatem triangvlarem eivs valor absolvtvs $\abs{L} + 1$ non excedit.
$\abs{f(x) + g(x)} \leq C_1\abs{h(x)} + C_2\abs{h(x)} = \resaltar{(C_1 + C_2)\abs{h(x)}}$
Margines $C_1$ ipsivs $f$ et $C_2$ ipsivs $g$ simvl valent prope $P$ , in intersectione dvarvm viciniarvm. Inaeqvalitas triangvlaris svmmam circvmscribit; mvltiplvm marginem $\abs{c}\,C_1$ habet.
$\abs{f_1(x) f_2(x)} \leq C_1\abs{g_1(x)} \cdot C_2\abs{g_2(x)} = \resaltar{C_1 C_2 \abs{g_1(x) g_2(x)}}$
Margines dvorvm factorvm mvltiplicantvr.
$0 \leq \abs{\frac{f_1(x) f_2(x)}{g_1(x) g_2(x)}} \leq C_1 \abs{\frac{f_2(x)}{g_2(x)}} \to \resaltar{0}$
Qvotiens prodvcti per $g_1 g_2$ est prodvctvm qvotientis per $C_1$ circvmscripti et alterivs qvi ad $0$ tendit; per theorema compressionis ad $0$ tendit.