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Regvla Hospitalii

Svmmarivm

Haec lectio regvlam Hospitalii demonstrat: si ff et gg ad 00 tendvnt, avt si gg ad infinitvm tendit, et qvotiens derivatarvm earvm limitem habet, tvm qvotiens fg\frac{f}{g} evndem limitem habet. Demonstratio theorema valoris medii Cavchii, ex lectione svperiore, in intervallo adhibet cvivs alter terminvs fixvs est, alter variabilis; casvs in qvo denominator ad infinitvm tendit argvmentvm svbtilivs postvlat, qvod nihil a nvmeratore exigit. Regvla valet ab vno latere pvncti, in pvncto et in infinito, et plvries continvo adhiberi potest: sic compvtatvr limes ipsivs f(x)−f(a)(x−a)n\frac{f(x) - f(a)}{(x - a)^n}, cvm primae derivatae ipsivs ff in aa evanescvnt. Formae 0⋅∞0 \cdot \infty, ∞−∞\infty - \infty, 1∞1^\infty, 000^0 et ∞0\infty^0 ad qvotientem redigvntvr. Lectio etiam ostendit qvando regvla non adhibetvr avt nihil decernit, cvr ad limites per qvos derivatae compvtatae svnt non valeat, et qvomodo cvm infinitesimis aeqvivalentibvs conivngatvr.

Proposita discendi

  1. Regvlam Hospitalii pro formis 00\frac{0}{0} et ∞∞\frac{\infty}{\infty}, ab vno latere pvncti, in pvncto et in infinito, ex theoremate valoris medii Cavchii demonstrare.
  2. Limites formarvm indeterminatarvm compvtare eas ad qvotientem redigendo, et regvlam plvries adhibere, hypotheses eivs in vnaqvaqve applicatione examinando.
  3. Agnoscere qvando regvla non adhibetvr avt nihil decernit: limites qvi non svnt indeterminati, qvotientes derivatarvm sine limite, derivatas denominatoris qvae prope pvnctvm evanescvnt, et limites qvos compvtatio derivatarvm praesvpponit.
  4. Regvlam cvm infinitesimis aeqvivalentibvs et cvm algebra limitvm conivngere, et in qvoqve casv viam brevissimam eligere.

Forma nihilvm per nihilvm

Corollarivm de limite qvotientis, ex lectione de algebra limitvm, postvlat vt denominator limitem a nihilo diversvm habeat. Si nvmerator et denominator ambo ad 00 tendvnt, qvotiens ad qvemlibet nvmervm tendere potest, avt ad infinitvm, avt limitem non habere: 3xx\frac{3x}{x} ad 33 tendit, x2x\frac{x^2}{x} ad 00 tendit, et xsin⁡1xx\frac{x \sin \frac{1}{x}}{x} limitem in 00 non habet. Ideo dicitvr 00\frac{0}{0} esse formam indeterminatam: limes cvivsqve termini limitem qvotientis non determinat. Lectio de infinitesimis aeqvivalentibvs mvltos horvm limitvm solvit, pro vnoqvoqve factore aeqvivalentem ex tabvla sva svbstitvendo; sed differentia vt x−sin⁡xx - \sin x non est prodvctvm factorvm tabvlae, et termini eivs svbstitvi non possvnt.

Derivata aliam viam svggerit. Si f(a)=g(a)=0f(a) = g(a) = 0, si ff et gg in aa derivabiles svnt et si g′(a)≠0g'(a) \neq 0, nvmeratore et denominatore per x−ax - a divisis qvotiens in qvotientem qvotientivm incrementalivm convertitvr, et corollarivm de limite qvotientis dat:

f(x)g(x)=f(x)−f(a)x−ag(x)−g(a)x−a→f′(a)g′(a)\frac{f(x)}{g(x)} = \frac{\frac{f(x) - f(a)}{x - a}}{\frac{g(x) - g(a)}{x - a}} \to \frac{f'(a)}{g'(a)}

Saepe avtem etiam g′(a)g'(a) nihilvm est, avt ff et gg in aa definitae non svnt. Regvla Hospitalii pro valoribvs derivatarvm in aa limitem qvotientis earvm prope aa svbstitvit. Instrvmentvm est theorema valoris medii Cavchii, ex lectione de theorematis Fermatii, Rollii, Lagrangii et Cavchii: si ff et gg continvae svnt in [a,b]\icc{a}{b}, cvm a<ba < b, et derivabiles in qvoqve pvncto ipsivs (a,b)\ioo{a}{b}, et si g′(x)≠0g'(x) \neq 0 pro omni xx ipsivs (a,b)\ioo{a}{b}, tvm g(b)≠g(a)g(b) \neq g(a) et exstat cc in (a,b)\ioo{a}{b} ita vt f(b)−f(a)g(b)−g(a)=f′(c)g′(c)\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)}. Dissimiliter a dvabvs applicationibvs theorematis Lagrangii, qvae dvo pvncta diversa darent, theorema Cavchii vnvm idemqve pvnctvm nvmeratori et denominatori dat.

TheoremaRegvla Hospitalii pro forma nihilvm per nihilvm

Sint a<ba < b, et sint ff et gg fvnctiones derivabiles in qvoqve pvncto ipsivs (a,b)\ioo{a}{b}, cvm g′(x)≠0g'(x) \neq 0 pro omni xx ipsivs (a,b)\ioo{a}{b}. Ponamvs f(x)f(x) et g(x)g(x) ad 00 tendere cvm x→a+x \to a^+, et

lim⁡x→a+f′(x)g′(x)=L\lim_{x \to a^+} \frac{f'(x)}{g'(x)} = L

vbi LL est nvmervs realis, ∞\infty avt −∞-\infty. Tvm g(x)≠0g(x) \neq 0 pro omni xx ipsivs (a,b)\ioo{a}{b}, et

lim⁡x→a+f(x)g(x)=L\lim_{x \to a^+} \frac{f(x)}{g(x)} = L

Fvnctiones ff et gg valore 00 in aa complentvr, et theorema Cavchii in [a,x]\icc{a}{x} adhibetvr: qvotiens f(x)g(x)\frac{f(x)}{g(x)} est qvotiens incrementorvm, aeqvalis qvotienti derivatarvm in pvncto inter aa et xx sito. Figvra ostendit, cvm f(x)=sin⁡xf(x) = \sin x, g(x)=ex−1g(x) = e^x - 1 et a=0a = 0, pvncta (g(t),f(t))\left(g(t), f(t)\right): qvotiens f(x)g(x)\frac{f(x)}{g(x)} est declivitas chordae qvae originem cvm pvncto ipsi xx respondenti ivngit, et qvotiens derivatarvm in cc declivitas tangentis parallelae.

Demonstratio

  1. F(a)=G(a)=0,F′(t)=f′(t),G′(t)=g′(t)≠0,a<t<xF(a) = G(a) = 0, \qquad F'(t) = f'(t), \quad G'(t) = g'(t) \neq 0, \qquad a < t < x

    Sint FF et GG fvnctiones qvae valent ff et gg in (a,b)\ioo{a}{b} et 00 in aa. Sit xx ipsivs (a,b)\ioo{a}{b}. Ambae continvae svnt in [a,x]\icc{a}{x}: in aa, qvia limites earvm a dextra svnt 00; in ceteris pvnctis, per theorema qvod fvnctionem derivabilem continvam esse affirmat, ex lectione de derivata vt limite. In qvoqve pvncto ipsivs (a,x)\ioo{a}{x} derivabiles svnt, cvm derivatis ipsarvm ff et gg, per lemma de indole locali derivatae, ex lectione de regvlis derivandi.

  2. f(x)g(x)=F(x)−F(a)G(x)−G(a)=f′(c)g′(c),a<c<x\frac{f(x)}{g(x)} = \frac{F(x) - F(a)}{G(x) - G(a)} = \dato{c}{\frac{f'(c)}{g'(c)}}, \qquad a < c < x

    Per theorema valoris medii Cavchii in [a,x]\icc{a}{x}, G(x)≠G(a)G(x) \neq G(a), id est g(x)≠0g(x) \neq 0, et exstat cc inter aa et xx ita vt qvotiens incrementorvm sit qvotiens derivatarvm in cc.

  3. a<x<a+δ→∣f(x)g(x)−L∣=∣f′(c)g′(c)−L∣<εa < x < a + \delta \rightarrow \abs{\frac{f(x)}{g(x)} - L} = \abs{\frac{f'(c)}{g'(c)} - L} < \resaltar{\varepsilon}

    Sit LL realis et sit ε>0\varepsilon > 0. Per definitionem limitis a dextra, ex lectione de limitibvs lateralibvs, exstat δ>0\delta > 0, qvod minvs qvam b−ab - a vel aeqvale svmi potest, ita vt ∣f′(t)g′(t)−L∣<ε\abs{\frac{f'(t)}{g'(t)} - L} < \varepsilon si a<t<a+δa < t < a + \delta. Si a<x<a+δa < x < a + \delta, pvnctvm cc articvli secvndi condicioni a<c<x<a+δa < c < x < a + \delta satisfacit. Si L=∞L = \infty, idem argvmentvm cvm provocatione NN dat f(x)g(x)>N\frac{f(x)}{g(x)} > N; si L=−∞L = -\infty, dat f(x)g(x)<N\frac{f(x)}{g(x)} < N.

Non opvs est scire qvomodo cc ab xx pendeat, neqve an vnicvm sit: satis est id inter aa et xx iacere. Hypothesis g′≠0g' \neq 0 in toto intervallo (a,b)\ioo{a}{b} postvlatvr, qvia theorema Cavchii ea in qvoqve [a,x]\icc{a}{x} vtitvr; cvm limes solvm a valoribvs propinqvis pendeat, satis est intervallvm (a,b)\ioo{a}{b} invenire vbi valeat, qvamvis parvvm. Non satis est contra derivatam denominatoris solvm in pvnctis isolatis evanescere qvae ad pvnctvm accvmvlantvr, avt qvae non circvmscripta svnt si variabilis ad infinitvm tendit, vt problema solvtvm 5 ostendit. Si ff et gg in copia maiore definitae svnt, theorema ad restrictiones earvm ad (a,b)\ioo{a}{b} adhibetvr, qvae evndem limitem a dextra in aa habent.

Idem valet a sinistra, cvm intervallo (b,a)\ioo{b}{a}, b<ab < a, et x→a−x \to a^-: satis est theorema ad f(−t)f(-t) et g(−t)g(-t) prope −a-a a dextra adhibere, per propositionem de translatione et reflexione, ex lectione de limitibvs lateralibvs; regvla catenae signvm ambarvm derivatarvm mvtat nec qvotientem earvm alterat. Si hypotheses ab vtroqve latere ipsivs aa valent, in (a−r,a)\ioo{a - r}{a} et in (a,a+r)\ioo{a}{a + r}, theorema de limite et limitibvs lateralibvs, ex illa lectione, limitem dat cvm x→ax \to a (si LL infinitvs est, eodem argvmento de minore dvorvm radiorvm). Fvnctio in aa definita esse potest vel non; valor eivs ibi nihil confert.

ExemplvmTres applicationes continvae

Consideremvs lim⁡x→0x−sin⁡xx3\lim_{x \to 0} \frac{x - \sin x}{x^3}. Nvmerator et denominator ad 00 tendvnt, et derivata denominatoris, 3x23x^2, extra 00 non evanescit. Qvotiens derivatarvm est 1−cos⁡x3x2\frac{1 - \cos x}{3x^2}, rvrsvs formae 00\frac{0}{0}, cvm denominatore cvivs derivata est 6x6x, qvae ne ipsa qvidem extra 00 evanescit; et novvs qvotiens, sin⁡x6x\frac{\sin x}{6x}, ad 16\frac{1}{6} tendit per limitem fvndamentalem trigonometriae:

lim⁡x→0x−sin⁡xx3=lim⁡x→01−cos⁡x3x2=lim⁡x→0sin⁡x6x=16\lim_{x \to 0} \frac{x - \sin x}{x^3} = \lim_{x \to 0} \frac{1 - \cos x}{3x^2} = \lim_{x \to 0} \frac{\sin x}{6x} = \frac{1}{6}

Series aeqvalitatvm a dextra ad sinistram legitvr: vltimvs limes exstat, et regvla, bis adhibita, valorem eivs retrorsvm transmittit. Primvm signvm aeqvalitatis scribere anteqvam sciatvr secvndvm limitem exstare modvs tantvm est compvtationem notandi, qvi ivstificatvr cvm ea perfecta est. Secvndvs limes etiam ex tabvla aeqvivalentivm colligitvr, cvm 1−cos⁡x∼x221 - \cos x \sim \frac{x^2}{2}; primvs non.