# Indertiminate Forms

*I am a PhD student of mathematics. I have complete MS in math from the University of Pakistan and have been writing online since 2020.*

## Indeterminate Forms.

**Introduction:**

"Indeterminate" refers to a value that is not known. The indeterminate form of a mathematical expression indicates the impossibility to determine the original value even after the limits have been substituted. We are not able to solve several types of mathematical expressions in mathematics.

We refer to these expressions as indeterminate forms. When two functions are taken in ratio, the indeterminate form typically arises when both functions approach 0 in the limit. These situations are known as "indeterminate form 0/0." Similar to addition, subtraction, multiplication, and exponential operations, the indeterminate form can also be obtained.

**Definition:**

An algebraic combination of functions in an independent variable are commonly evaluated in calculus and other branches of mathematics by substituting these functions with their limits; if the expression obtained after this substitution does not provide enough information to determine the original limit, the expression is called indeterminate form.

**Types of Indeterminate Form:**

When given limit is applied on the expression sometimes it does not provide enough information, this expression is referred to indeterminate form. There are seven types of indeterminate forms like,

** **

**1. ****0 / 0 form (Zero by Zero Form)**

Zero by Zero form (0 / 0) is obtained when Lim x→ 0 is applied on the fractional factor like x/ * x^{3}*, 2x / x., √x / x, x/ the resulting expression in each example is 0 / 0 form which is indeterminate form if the limits of the numerator and denominator are substituted.

**2. ****∞ / ∞ Form (Infinity by Infinity Form):**

Infinity by infinity form is obtained when Lim x→ ∞ is applied on the quotient factor like, **x / x^{3}**, 2x / x., √x / x, x/ the resulting expression in each example is

**∞**/

**∞**form which is indeterminate form if the limits of the numerator and denominator are substituted.

**3. **** (zero Multiply by Infinity):**

Zero Multiply by Infinity is obtained when Lim x→ 0 is applied on the factor like, x,2x, 3x,* **x ^{n}* and √x, √

*x*where n is any real number as well as Lim x→ ∞ is applied on the factor like, x,2x, 3x,

^{n}

*x*and √x, √

^{n}

*x*

^{n}**4. ****∞ - **** ****∞ (Infinity minus Infinity):**

Infinity minus Infinity is obtained when Lim x→ ∞ is applied on the factor like, x – x, 2x – 4x,* **x ^{m}* -

*x*, √

^{n}

*x*-

^{m}*x*where m and n are equal different, and m and n are not equal to zero.

^{n}**5. 1****^∞ (x power infinity):**

x power infinity is obtained when limit Lim x→ ∞ is applied on the factor like,* **1^*√* **x ^{x}*

*1*

*^*

*x*

^{n}*1*, where n is any real number.

^{x+n}1^{x}

*6. ***0 ^{0} (Zero power Zero)**

*Zero power Zero is obtained when *Lim x→ 0 is applied on the factor in some power of zero like, 0^{x}* **0 ^*√* **x, ** **0 ^{x+1}* , etc.0

^{x}

**7. ****∞^{0}**

**(Infinity Power Zero):**

Infinity Power Zero is obtained by applied the limit x→ 0 on the factor of some power of infinity like, ∞* ^{x}* ∞

*where n is any real number.*

^{x+n}**Examples of Indeterminate Form:**

Example No.1:

Given that:

Lim x→ 2 ((x*^{2}*-4) / (x-2))

Solution:

Apply the Lim x→ 2 on the factor (x*^{2}*-4) / (x-2) it gives,

By quotient rule of limit,

Lim x→ 2 (x*^{2}*-4) / Lim x→ 2(x-2)

Now according to subtraction rule of limit,

Lim x→ (x*^{2}*- Lim x→ 2(4) / Lim x→ 2 (x)- Lim x→ 2(2)

4-4 \ 2-2

0 / 0

Which is indeterminate form of Lim x→ 2 ((x*^{2}*-4) / (x-2))

Now we will vanish the indeterminate form by factorization.

Lim x→ 2 ((x*^{2}*-4) / (x-2))

Lim x→ 2 (x+2) (x-2) / Lim x→ 2 (x-2)

According to multiplication rule of limit,

Lim x→ 2(x+2). Lim x→ 2 (x-2) / Lim x→ 2 (x-2)

By cancellation law it gives,

Lim x→ 2(x+2)

Now apply the limit on the factor, (x+2) it gives,

2+2 = 4

Hence,

Lim x→ 2 ((x*^{2}*-4) / (x-2)) = 4

**Example No.2: **

**Given That:**

**Lim x→ ****∞ ((x**^{2}**/ x)) ….. (1)**

**Solution:**

When apply the limit on equation no.1 it gives indeterminate form of for vanish the indeterminate form, we apply the method of factorization.

Lim x→ ∞ ((x^{2}/ x)) = Lim x→ ∞(x(x) / x)

By cancellation principle,

Lim x→ ∞ ((x^{2}/ x)) = Lim x→ ∞ (x)

⇒ ∞

∞ is not an indeterminate form.

Hence,

Lim x→ ∞ ((x^{2}/ x)) = ∞

**FAQs**

**1. What does it mean when a form is indeterminate?**

An algebraic combination of functions in an independent variable are commonly evaluated in calculus and other branches of mathematics by substituting these functions with their limits; if the expression obtained after this substitution does not provide enough information to determine the original limit, the expression is called indeterminate form.

**2. Will when 0 / 0 form (Zero by Zero Form) occur?**

Zero by Zero form (0 / 0) is obtained when Lim x→ 0 is applied on the fractional factor like x/ * x^{3}* x/ x, 2x / x., √x / x, the resulting expression in each example is 0 / 0 form which is indeterminate form if the limits of the numerator and denominator are substituted.

**3. Will when ∞ - ∞ (Infinity minus Infinity):**

Infinity minus Infinity is obtained when Lim x→ ∞ is applied on the factor like, x – x, 2x – 4x,* ** x^{m}* -

*, √*

**x**^{n}

*-*

**x**^{m}*where m and n are equal different, and m and n are not equal to zero.*

**x**^{n}**4. Give any example of indeterminate form?**

**Given That:**

**Lim x→ ∞ ((x**^{2}**/ x)) ….. (1)**

**Solution:**

When apply the limit on equation no.1 it gives indeterminate form of for vanish the indeterminate form, we apply the method of factorization.

Lim x→ ∞ ((x^{2}/ x)) = Lim x→ ∞(x(x) / x)

By cancellation principle,

Lim x→ ∞ ((x^{2}/ x)) = Lim x→ ∞ (x)

⇒ ∞

∞ is not an indeterminate form.

Hence,

Lim x→ ∞ ((x^{2}/ x)) = ∞

**5. What is importance of indeterminate form in calculus?**

When evaluating the limits of function, indeterminate forms are frequently encountered, and limits in turn have a significant impact on mathematics and calculus. For understanding about gradients, derivatives, and many other topics, they play an important part in calculus.

**Related Links:**

- Limit of a Function in Mathematics.

Limits plays an important part in calculus as well as in our daily life. The concept of continuity, derivative and integral is based on limits. It is a most basic concept of calculus.

- Concept of Infinity in Mathematics

Infinity is most important term which is used in every field of life as well as every field of science and arts. Everything which is unbounded refer to infinity.

- L’Hospital’s Rule in Calculus

The L'Hospital's rule is the most famous mathematical tool among students because of its effectiveness and simplicity. By L'Hospital's rule we can solve the intermediate forms like 0/0 or ∞/∞ etc.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2022 Kinza Javaid**