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Unbestimmte Formen

In: Vorlesungen über höhere Mathematik

Author

Listed:
  • Adalbert Duschek

    (Technischen Hochschule Wien)

Abstract

Zusammenfassung Die Regel für den Grenzwert eines Quotienten % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaWGHbaabeaakmaa % laaabaGaamOzaiaacIcacaWG4bGaaiykaaqaaiaadEgacaGGOaGaam % iEaiaacMcaaaGaeyypa0ZaaSaaaeaadaWfqaqaaiGacYgacaGGPbGa % aiyBaiaaygW7caWGMbGaaiikaiaadIhacaGGPaaaleaacaWG4bGaey % OKH4QaamyyaaqabaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaiaa % ygW7caWGNbGaaiikaiaadIhacaGGPaaaleaacaWG4bGaeyOKH4Qaam % yyaaqabaaaaaaa!5BD3! $$\mathop {\lim }\limits_{x \to a} \frac{{f(x)}}{{g(x)}} = \frac{{\mathop {\lim f(x)}\limits_{x \to a} }}{{\mathop {\lim g(x)}\limits_{x \to a} }}$$ von § 7, 10 gilt nur, wenn der Grenzwert des Nenners % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaWGHbaabeaakiaa % dEgacaGGOaGaamiEaiaacMcacqGHGjsUcaaIWaaaaa!429A! $$\mathop {\lim }\limits_{x \to a} g(x) \ne 0$$ ist. Damit ist aber nicht gesagt, daß der Quotient überhaupt keinen Grenzwert besitzt, wenn % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaWGHbaabeaakiaa % dEgacaGGOaGaamiEaiaacMcacqGHGjsUcaaIWaaaaa!429A! $$\mathop {\lim }\limits_{x \to a} g(x) \ne 0$$ ist, wie das Beispiel % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIWaaabeaakmaa % laaabaGaci4CaiaacMgacaGGUbGaaGzaVlaadIhaaeaacaWG4baaai % abg2da9iaadMeaaaa!44EB! $$\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} = I$$ zeigt. Ist z. B. gleichzeitig auch % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaWGHbaabeaakiaa % dEgacaGGOaGaamiEaiaacMcacqGHGjsUcaaIWaaaaa!429A! $$\mathop {\lim }\limits_{x \to a} g(x) \ne 0$$ , so kann der Quotient sehr wohl einen bestimmten (eigentlichen oder uneigentlichen) Grenzwert haben. Ich beweise zunächst den folgenden Satz:

Suggested Citation

  • Adalbert Duschek, 1956. "Unbestimmte Formen," Springer Books, in: Vorlesungen über höhere Mathematik, edition 0, chapter 0, pages 220-227, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7091-3556-3_21
    DOI: 10.1007/978-3-7091-3556-3_21
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