Name: limits of zero times infinity
Uploaded: 2026-01-12T12:24:55.000Z
Duration: 1 min 29 s
Channel: JoVE

Unbestimmte Formen können in Grenzwerten von Produkten entstehen, wobei ein Faktor gegen null geht und der andere zu plus oder minus unendlich tendiert.Das Ergebnis ist unbestimmt oder unklar. Es könnte null, unendlich oder eine endliche Zahl sein.Betrachten wir zum Beispiel den einseitigen Grenzwert der folgenden Produktfunktion.Wenn x auf null zuneigt, sinkt der erste Faktor x auf null, während der zweite Faktor, das natürliche Logarithmus von eins über x, gegen unendlich neigt.Das Produkt ist algebraisch gleich einem Quotient, der in zwei verschiedenen Ausdrücken geschrieben werden kann.Der erste Ausdruck erzeugt eine Unendlichkeit über Unendlichkeit-Form, während der zweite Ausdruck eine Null-über-Null-Form erzeugt.Die Regel von L'Hôpital kann auf beide Formen angewendet werden. Die Verwendung der ersten Form vereinfacht jedoch die Differenzierung und ergibt als Ergebnis Null.Eine hilfreiche Visualisierung ist ein regelmäßiges Polygon, das in einen Kreis eingraviert ist. Wenn die Anzahl der Seiten unendlich zunimmt, schrumpft jede Seite auf null zu.Dies erzeugt ein unbestimmtes Produkt: unendliche Seiten mal null Länge. Dennoch nähert sich der gesamte Umfang dem festen Umfang des Kreises.Dies zeigt, wie eine null-mal-unendliche Form ein sinnvolles, endliches Ergebnis erzeugen kann.

unbestimmte produkte

Indeterminate Products

Unbestimmte Formen treten auch bei der Berechnung von Grenzwerten bei Produkten auf, insbesondere wenn ein Faktor gegen null strebt, w&#xE4;hrend der andere gegen positive oder negative Unendlichkeit strebt. Diese Situation, &#xFC;blicherweise als Null-mal-Unendlich-Form bezeichnet, liefert kein unmittelbar interpretierbares Ergebnis. Abh&#xE4;ngig davon, wie sich die Faktoren zueinander verhalten, kann der Grenzwert eines solchen Produkts null, unendlich oder ein von null verschiedener endlicher Wert sein.
Grenzwerte von Produkten und algebraische Umformung
Um Grenzwerte dieser Art zu analysieren, muss das Produkt zun&#xE4;chst als Quotient umgeschrieben werden, um die Regel von L&#x2019;H&#xF4;pital anwenden zu k&#xF6;nnen. Betrachtet wird ein einseitiger Grenzwert, bei dem die Variable gegen null strebt, wodurch ein Faktor gegen null abnimmt, w&#xE4;hrend der andere Faktor, beispielsweise der nat&#xFC;rliche Logarithmus des Kehrwerts der Variablen, gegen unendlich w&#xE4;chst. Obwohl der urspr&#xFC;ngliche Ausdruck ein Produkt ist, kann er algebraisch in zwei &#xE4;quivalente Quotientenformen umgeformt werden. Eine Umformung liefert die Form PT_19834_1, die andere die Form PT_19834_2. Beide Darstellungen sind g&#xFC;ltig und erlauben die Anwendung der Regel von L&#x2019;H&#xF4;pital.
Die Anwendung der Regel umfasst das separate Ableiten von Z&#xE4;hler und Nenner, gefolgt von der Berechnung des resultierenden Grenzwerts. Zwar kann jede der Quotientenformen verwendet werden, doch die Wahl der Form beeinflusst die Einfachheit der Ableitung. In diesem Fall ergibt die Unendlich-zu-Unendlich-Form eine einfachere Ableitung, wodurch sich zeigt, dass der Grenzwert des urspr&#xFC;nglichen Produkts null ist.
Geometrische Interpretation
Eine n&#xFC;tzliche geometrische Analogie zum Verst&#xE4;ndnis von Null-mal-Unendlich-Formen ist ein regelm&#xE4;&#xDF;iges Polygon, das in einen Kreis eingeschrieben ist. Wenn die Anzahl der Seiten des Polygons unbegrenzt w&#xE4;chst, nimmt die L&#xE4;nge jeder einzelnen Seite gegen null ab. Der Gesamtumfang ist jedoch das Produkt aus der Anzahl der Seiten und der L&#xE4;nge jeder Seite. Dies f&#xFC;hrt zu einem unbestimmten Produkt mit unendlich vielen Seiten gegen null gehender L&#xE4;nge. Dennoch n&#xE4;hert sich der Umfang einem endlichen, wohldefinierten Wert, der dem Kreisumfang entspricht. Dieses Beispiel verdeutlicht, dass Null-mal-Unendlich-Formen bei geeigneter Analyse zu sinnvollen, endlichen Grenzwerten f&#xFC;hren k&#xF6;nnen.

Unbestimmte Produkte

Unbestimmte Formen treten auch bei der Berechnung von Grenzwerten bei Produkten auf, insbesondere wenn ein Faktor gegen null strebt, w&#xE4;hrend der ...

Education

Lehre

JoVE Core

Calculus

Analysis

Unit

Einheit

Applications of Differentiation

Anwendungen der Differenzialrechnung

KEY TERMS AND CONCEPTS

SCHLÜSSELBEGRIFFE UND KONZEPTE

limits of zero times infinity

Limits with a zero-times-infinity product are indeterminate. In this type of limit, one factor approaches zero while the other factor grows toward positive or negative infinity. The result is not obvious at first, because the product may approach zero, infinity, or a finite nonzero value.

To evaluate these product limits, the expression is first rewritten as a quotient. This form makes it possible to use L'Hôpital's Rule, which compares the derivatives of the numerator and denominator. A one-sided limit can be rewritten in more than one equivalent quotient form. Either form is valid, but one may be easier to differentiate than the other.

In the example shown, the form with infinity over infinity gives a simpler derivative. After applying L'Hôpital's Rule, the limit of the original product is zero. This shows that a zero-times-infinity expression can have a definite limit once it is rewritten and analyzed correctly.

A geometric example helps make the idea clearer. A regular polygon inscribed in a circle has sides that get shorter as the number of sides increases without bound. At the same time, the number of sides grows without bound, so the perimeter is a product of many sides and very small side lengths. Even though this is an indeterminate product, the perimeter approaches the circle's circumference, which is finite and well defined.

Limits of Zero Times Infinity

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Limits with a zero-times-infinity product are indeterminate. In this type of limit, one factor approaches zero while the other factor grows toward positive or negative infinity. The result is not obvious at first, because the product may approach zero, infinity, or a finite nonzero value. To evaluate these product limits, the expression is first rewritten as a quotient. This form makes it possible to use L'Hôpital's Rule, which compares the derivatives of the numerator and denominator. A...

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key terms and concepts

Learn how zero-times-infinity limits work, why they are indeterminate, and how rewriting the product as a quotient helps apply L'Hôpital's Rule.

Limits of Zero Times Infinity | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Limits of Zero Times Infinity

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