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

Belirsiz biçimler, çarpımların limitlerinde ortaya çıkabilir; burada bir faktör sıfıra yaklaşırken, diğeri sonsuzluk artı ya da eksi eğilimli olur.Sonuç belirsiz veya belirsizdir. Sıfır, sonsuz veya sonlu sayı olabilir.Örneğin, aşağıdaki çarpım fonksiyonunun tek taraflı sınırını ele alalım.x sıfıra eğilimlendikçe, birinci faktör x sıfıra düşerken, ikinci faktör, x üzerinde birin doğal log'u sonsuzluğa doğru eğilimlidir.Bu çarpım, cebirsel olarak iki farklı ifadede yazılabilen bir bölüme eşittir.İlk ifade sonsuzluk üzerinde bir form oluştururken, ikinci ifade sıfır üzerinde sıfır formu oluşturur.L'Hôpital'ın Kuralı her iki forma da uygulanabilir. Ancak, birinci formu kullanmak farklılaştırmayı basitleştirir ve bunun sonucunda sıfır verir.Yardımcı bir görselleştirme, daire içinde kazınmış düzenli bir çokgendir. Kenar sayısı sonsuz arttıkça, her iki taraf sıfıra doğru küçülür.Bu, belirsiz bir çarpım oluşturur: sonsuz kenarlar çarpı sıfır uzunluk. Ancak, toplam çevre dairenin sabit çevresine yaklaşır.Bu, sıfır çarp sonsuzluk biçiminin anlamlı, sonlu bir sonuç nasıl üretebileceğini gösterir.

belirsiz Çarpımlar

Indeterminate Products

Belirsiz bi&#xE7;imler, &#xF6;zellikle bir &#xE7;arpan&#x131;n s&#x131;f&#x131;ra yakla&#x15F;&#x131;rken di&#x11F;er &#xE7;arpan&#x131;n art&#x131; ya da eksi sonsuza y&#xF6;neldi&#x11F;i durumlarda, &#xE7;arp&#x131;m i&#xE7;eren limitlerin de&#x11F;erlendirilmesinde de ortaya &#xE7;&#x131;kar. Genellikle s&#x131;f&#x131;r &#xE7;arp&#x131; sonsuz bi&#xE7;imi olarak adland&#x131;r&#x131;lan bu durum, do&#x11F;rudan yorumlanabilir bir sonuca sahip de&#x11F;ildir. &#xC7;arpanlar&#x131;n g&#xF6;reli davran&#x131;&#x15F;lar&#x131;na ba&#x11F;l&#x131; olarak, bu t&#xFC;r bir &#xE7;arp&#x131;m&#x131;n limiti s&#x131;f&#x131;r, sonsuz ya da s&#x131;f&#x131;rdan farkl&#x131; sonlu bir de&#x11F;er olabilir.
&#xC7;arp&#x131;m Limitleri ve Cebirsel Yeniden Yazma
Bu t&#xFC;r limitleri analiz edebilmek i&#xE7;in, L&#x2019;H&#xF4;pital Kural&#x131;&#x2019;n&#x131;n uygulanmas&#x131;na olanak tan&#x131;yacak &#x15F;ekilde &#xE7;arp&#x131;m&#x131;n &#xF6;ncelikle bir oran olarak yeniden yaz&#x131;lmas&#x131; gerekir. De&#x11F;i&#x15F;kenin s&#x131;f&#x131;ra yakla&#x15F;t&#x131;&#x11F;&#x131;, bir &#xE7;arpan&#x131;n s&#x131;f&#x131;ra do&#x11F;ru azal&#x131;rken di&#x11F;er &#xE7;arpan&#x131;n (&#xF6;rne&#x11F;in de&#x11F;i&#x15F;kenin tersinin do&#x11F;al logaritmas&#x131; gibi) s&#x131;n&#x131;rs&#x131;z bi&#xE7;imde artt&#x131;&#x11F;&#x131; tek tarafl&#x131; bir limiti ele alal&#x131;m. Ba&#x15F;lang&#x131;&#xE7;taki ifade bir &#xE7;arp&#x131;m olmas&#x131;na ra&#x11F;men, cebirsel olarak birbirine e&#x15F;de&#x11F;er iki oran bi&#xE7;iminde yeniden d&#xFC;zenlenebilir. Bu d&#xFC;zenlemelerden biri PT_19834_1 bi&#xE7;imini, di&#x11F;eri ise PT_19834_2 bi&#xE7;imini verir. Her iki g&#xF6;sterim de ge&#xE7;erlidir ve L&#x2019;H&#xF4;pital Kural&#x131;&#x2019;n&#x131;n uygulanmas&#x131;na olanak tan&#x131;r.
Kural&#x131;n uygulanmas&#x131;, pay&#x131;n ve paydan&#x131;n ayr&#x131; ayr&#x131; t&#xFC;rev al&#x131;nmas&#x131;n&#x131; ve ard&#x131;ndan elde edilen ifadenin limitinin de&#x11F;erlendirilmesini i&#xE7;erir. Her iki oran bi&#xE7;imi de kullan&#x131;labilmekle birlikte, tercih edilen bi&#xE7;im t&#xFC;rev alma i&#x15F;leminin kolayl&#x131;&#x11F;&#x131;n&#x131; etkileyebilir. Bu durumda, sonsuz/sonsuz bi&#xE7;imi daha sade bir t&#xFC;rev ifadesi verir ve ba&#x15F;lang&#x131;&#xE7;taki &#xE7;arp&#x131;m&#x131;n limitinin s&#x131;f&#x131;r oldu&#x11F;unu ortaya koyar.
Geometrik Yorum
S&#x131;f&#x131;r &#xE7;arp&#x131; sonsuz belirsiz bi&#xE7;imlerini anlamaya y&#xF6;nelik yararl&#x131; bir geometrik benzetme, bir &#xE7;ember i&#xE7;ine &#xE7;izilmi&#x15F; d&#xFC;zg&#xFC;n bir &#xE7;okgendir. &#xC7;okgenin kenar say&#x131;s&#x131; s&#x131;n&#x131;rs&#x131;z bi&#xE7;imde artt&#x131;k&#xE7;a, her bir kenar&#x131;n uzunlu&#x11F;u s&#x131;f&#x131;ra yakla&#x15F;&#x131;r. Buna kar&#x15F;&#x131;l&#x131;k, toplam &#xE7;evre, kenar say&#x131;s&#x131; ile her bir kenar&#x131;n uzunlu&#x11F;unun &#xE7;arp&#x131;m&#x131;d&#x131;r. Bu durum, uzunluklar&#x131; s&#x131;f&#x131;ra yakla&#x15F;an sonsuz say&#x131;da kenar&#x131; i&#xE7;eren belirsiz bir &#xE7;arp&#x131;m olu&#x15F;turur. Buna ra&#x11F;men, &#xE7;okgenin &#xE7;evresi &#xE7;emberin &#xE7;evresine e&#x15F;it olan sonlu ve iyi tan&#x131;mlanm&#x131;&#x15F; bir de&#x11F;ere yakla&#x15F;&#x131;r. Bu &#xF6;rnek, s&#x131;f&#x131;r &#xE7;arp&#x131; sonsuz bi&#xE7;imlerinin uygun &#x15F;ekilde analiz edildi&#x11F;inde anlaml&#x131; ve sonlu limitler verebilece&#x11F;ini g&#xF6;stermektedir.

Belirsiz Çarpımlar

Belirsiz bi&#xE7;imler, &#xF6;zellikle bir &#xE7;arpan&#x131;n s&#x131;f&#x131;ra yakla&#x15F;&#x131;rken di&#x11F;er &#xE7;arpan&#x131;n art&#x131; ya da ...

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Applications of Differentiation

Türevin Uygulamaları

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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Using Second Derivatives to Read Curvature

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optimization problems

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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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