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Indeterminate Products

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Indeterminate forms can arise in limits of products, where one factor approaches zero, and the other tends to plus or minus infinity.

The result is indeterminate or unclear. It could be zero, infinite, or a finite number.

For example, consider the one-sided limit of the following product function.

As x tends to zero, the first factor x decreases to zero, while the second factor, the natural log of one over x, tends to infinity.

The product is algebraically equal to a quotient that can be written in two different expressions.

The first expression creates an infinity over infinity form, while the second expression creates a zero over zero form.

L’Hôpital’s Rule can be applied to either form. But, using the first form simplifies the differentiation and gives zero as a result.

A helpful visualization is a regular polygon inscribed in a circle. As the number of sides increases infinitely, each side shrinks toward zero.

This creates an indeterminate product: infinite sides times zero length. Yet, the total perimeter approaches the circle’s fixed circumference.

This shows how a zero-times-infinity form can produce a meaningful, finite result.

Overview

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$$\rightleftharpoonup{xx}$$ $$\longleftharp{xx}$$, $$\longrightharp{xx}$$,

Indeterminate forms also arise in the evaluation of limits involving products, particularly when one factor approaches zero while the other tends to positive or negative infinity. This situation, commonly described as a zero-times-infinity form, does not have an immediately interpretable outcome. Depending on how the factors behave relative to one another, the limit of such a product may be zero, infinite, or a finite nonzero value.

Product Limits and Algebraic Rewriting

To analyze limits of this type, the product must first be rewritten as a quotient so that L’Hôpital’s Rule can be applied. Consider a one-sided limit in which the variable approaches zero, causing one factor to decrease toward zero while the other factor, such as the natural logarithm of the reciprocal of the variable, increases without bound. Although the original expression is a product, it can be rewritten algebraically in two equivalent quotient forms. One rearrangement produces a Indeterminate form 0/0 symbol, mathematical concept, limits evaluation, calculus study. form, while the other produces a Infinity divided by infinity symbol, mathematical concept, equation, diagram, limits analysis. form. Both representations are valid and allow the use of L’Hôpital’s Rule.

Applying the rule involves differentiating the numerator and denominator separately and then evaluating the resulting limit. While either quotient form may be used, the choice of form can affect the simplicity of the differentiation. In this case, the form of infinity over infinity yields a simpler derivative, revealing that the limit of the original product is zero.

Geometric Interpretation

A useful geometric analogy for understanding zero-times-infinity indeterminate forms is a regular polygon inscribed in a circle. As the number of sides of the polygon increases without bound, the length of each individual side decreases toward zero. The total perimeter, however, is the product of the number of sides and the length of each side. This produces an indeterminate product involving infinitely many sides of vanishing length. Despite this, the perimeter approaches a finite, well-defined value equal to the circumference of the circle. This example illustrates how zero-times-infinity forms can yield meaningful finite limits when analyzed appropriately.

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Indeterminate Forms

Zero Times Infinity

L H pital s Rule

Algebraic Rewriting

Geometric Interpretation