Indeterminate Forms: List & Calculation

Indeterminate form is a mathematical phrase that states that even after substituting the limits, we cannot determine the original value. In most cases, it involves two fractions whose limits cannot be established by referring to the initial limits of the two functions separately. 

• The indeterminate form is a branch of calculus that includes seven expressions, namely 0/0, 0^∞, ∞/∞, 0 x ∞, 0⁰, 1^∞ and ∞ – ∞.
• The term was first introduced by Cauchy's student Moigno in the 19th century.
• It is used to solve the limit of the sum, difference, product, quotient and power of two given functions.
• The phrase "indeterminate" refers to a value that is unknown.
• In this case, the limit of the derivative tends to become the limit of an indeterminate form.

Key Terms: Indeterminate Forms, Indeterminate Forms of Limits, Factoring Method, Indeterminate Forms List, L’Hospital’s Rule, Division of Each Term by Highest Power of Variable

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What is Indeterminate Form?

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Indeterminate forms are expressions that are unable to find answers to some equations in mathematics. In most circumstances, the indeterminate form occurs when a ratio of two functions is computed so that both functions approach zero in the limit. 

• Such situations are referred to as "indeterminate form 0/0." 
• In addition, subtraction, multiplication, and exponential operations can all be used to obtain the required form.
• It is a combination of a maximum of two values of 0, 1 and ∞.
• Indeterminate forms are classified as "indeterminate" as it help distinguish other ratios that may be 0 or do not exist. 
• It's possible that replacing the word "indeterminate" with "temporary" will clear up some of the misunderstandings around the use of such forms. 

As a result, an indeterminate form might be viewed as merely a tool for additional calculation, which is left to the discretion of the information consumer.

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Indeterminate Forms of Limits

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If we delve into the complexities of indeterminate forms, we can see how they may be used in a variety of ways to determine the features of any function at any given limit. 

• Indeterminate forms are frequently solvable; they can determine value that was previously unknown.
• Zero in the denominator could indicate infinity or the absence of anything. 
• There is occasionally too much data that necessitates reducing an answer to a specific technique. 
• In order to solve for a limit when working with proportions like 1/0 and 0/0 or with an eternity of any type, we will almost certainly need to utilize another theorem, such as L'Hopital's Rule. 

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Indeterminate Forms List

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There are seven indeterminate forms with 0/0 and ∞/∞ are the most occurring forms. The detailed analysis of indeterminate forms list are as follows:

Indeterminate Form 1

The first form is 0/0

Condition: The required condition include

lim_x--->c f(x) = 0 and lim_x--->c g(x) = 0

Transformation: Transformation of indeterminate of the form ∞/∞ will be given as: lim_x--->c f(x) /g(x)

lim_x--->c 1/f(x)/1/g(x) 

Indeterminate Form 2

The second form is ∞/∞

Condition: The required condition include

lim_x--->c f(x) = ∞ and lim_x--->c g(x) = ∞

Transformation: Transformation of indeterminate of the form ∞/∞ will be given as: lim_x--->c f(x) /g(x)

lim_x--->c 1/f(x)/1/g(x) 

Indeterminate Form 3

The third form is 0 x ∞

Condition: The required condition include

lim_x--->c f(x) = 0 and lim_x--->c g(x) = ∞

Transformation: Transformation of indeterminate of the form ∞/∞ will be divided into 0/0 and ∞/∞.

• First lim_x--->c f(x)g(x) and then into lim_x--->c f(x)/1/g(x) 
• Second lim_x--->c f(x)g(x) and then into lim_x--->c g(x)/1/f(x) 

Indeterminate Form 4

The fourth form is 1^∞

Condition: The required condition include

lim_x--->c f(x) = 1 and lim_x--->c g(x) = ∞

Transformation: Transformation of indeterminate of the form ∞/∞ will be divided into 0/0 and ∞/∞.

• First lim_x--->c f(x)^g(x) and then into lim_x--->c lnf(x)/1/g(x) 
• Second lim_x--->c f(x)^g(x) and then into lim_x--->c g(x)/1/lnf(x) 

Indeterminate Form 5

The fifth form is 0⁰

Condition: The required condition include

lim_x--->c f(x) = 0⁺ and lim_x--->c g(x) = 0

Transformation: Transformation of indeterminate of the form ∞/∞ will be divided into 0/0 and ∞/∞.

• First lim_x--->c f(x)^g(x) and then into lim_x--->c g(x)/1/lnf(x) 
• Second lim_x--->c f(x)^g(x) and then into lim_x--->c lnf(x)/1/g(x) 

Indeterminate Form 6

The sixth form is ∞⁰

Condition: The required condition include

lim_x--->c f(x) = ∞ and lim_x--->c g(x) = 0

Transformation: Transformation of indeterminate of the form ∞/∞ will be divided into 0/0 and ∞/∞.

• First lim_x--->c f(x)^g(x) and then into lim_x--->c g(x)/1/lnf(x) 
• Second lim_x--->c f(x)^g(x) and then into lim_x--->c lnf(x)/1/g(x) 

Indeterminate Form 7

The seventh form is ∞ – ∞

Condition: The required condition include

lim_x--->c f(x) = ∞ and lim_x--->c g(x) = ∞

Transformation: Transformation of indeterminate of the form ∞/∞ will be divided into 0/0 and ∞/∞.

• First lim_x--->c f(x) – g(x) and then into lim_x--->c [(1/g(x) – 1/lnf(x)) / 1/f(x)g(x)}
• Second lim_x--->c f(x) – g(x) and then into lim_x--->c e^f(x)/e^g(x) 

Indeterminate Forms List 

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How to evaluate Indeterminate Forms?

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Different ways to evaluate indeterminate forms are as follows:

Factoring Method

Factoring Method is normally used for 0/0 form, and it entails factoring the given equations to their simplest terms. The limit value is utilized to solve after the simplest form has been derived.

L'Hospital's Rule

In the case of an indeterminate form, the L'Hospital's rule states that the best way to solve it is to differentiate the numerator and denominator independently before applying the limit.

• After each stage, the derivatives of the numerator and denominator are examined individually to check if they have become free of the variable.
• This results in at least one of the terms remaining constant.

Division by Highest Power

When the undetermined form is normally presented in the / format, division by highest power technique is usually used. In this scenario, the best course of action is to divide both the numerator and denominator of the supplied expression by the highest power variable in the sum. Following that, the limit value is determined.

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Indeterminate Forms Example

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The indeterminate forms example are as follows:

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Things to Remember

• Indeterminate form indicates that, even after the limitations have been substituted, we are still unable to ascertain the original value.
• Four algebraic procedures are used to evaluate limits of indeterminate forms.
• These procedures are sometimes referred to as factorization (fraction reduction), rationalization, or trigonometric rules in textbooks.
• Many researchers use the derivative approach to solve indeterminate forms.
• An indeterminate form does not imply that the limit does not exist or cannot be identified, but rather that its limits' attributes are invalid.