Not everything is defined in maths. In fact, there are some regions in maths which we can't even imagine. Anything that can't be defined by maths is called an indeterminate form. An indeterminate form does not mean that the limit is non-existent or cannot be determined, but rather that the properties of its limits are not valid. In these cases, a particular operation can be performed to solve each of the indeterminate forms. In simple words, you need to have a different approach to handle such problems, many mathematicians use the differentiation method to solve the indeterminate form but this is not the only method, there are several methods which you can use but let's just stick to the indeterminate forms with its types for now.

Indeterminate Form

Indeterminate form comes in various shapes. To understand the indeterminate form, it is important to learn about its types.

1. Infinity over Infinity

For example, you are given a function, \lim_{ x \rightarrow \infty } \frac { \frac { 1 }{ x } + 2 }{ { 2 }{ 5x } - 5 }. After applying limits, you will get \frac { \infty }{ \infty } which can't be solved. There is no proper solution of this fraction and that is why we can conclude it as an indeterminate form.

\frac { \infty }{ \infty } \rightarrow Ind

2. Infinity Minus Infinity

When you subtract infinity from infinity. Again there is ambiguity in the equation. You cannot minus infinity from infinity, we can't find a proper outcome. Hence, it is considered an indeterminate form.

\infty - \infty \rightarrow Ind

3. Zero over Zero

One of the most common indeterminate examples is zero over zero. Dividing any number by zero is undefined, it could be any value. The reason is that the division will never be completed. You keep dividing the numerator with zero and it will keep going till infinity. Therefore, zero over zero is a very common indeterminate form.

\frac { 0 }{ 0 } \rightarrow Ind

4. Zero Times Infinity

We talked about infinity over infinity, and zero over zero, what about zero times infinity? The answer is undefined again! It could be any number that we can't predict. Many people make this mistake, they think that the answer is zero because anything multiplied by zero is zero but what they don't realize is the infinity sign with it.

0 \times \infty \rightarrow Ind

5. Zero to the Power of Zero

This problem is similar to the division by zero. Mathematics rule says that any positive number, besides zero, whose power is equal to zero will be equal to one. So, it means { 0 }^{ 0 } is 1 but how can a zero entity be equal to one? It is impossible for zero to become one or any other number at any cost. Hence, it is undefined and we can call it an indeterminate form.

{ 0 }^{ 0 } \rightarrow Ind

6. Infinity to the Power of Zero

Infinity value doesn't have a universal value. Infinity having a power equal to zero is also undefined hence it is also a type of indeterminate form.

{ \infty }^{ 0 } \rightarrow Ind

7. One to the Power of Infinity

Last but not least, one to the power infinity is also a type of indeterminate form. Since we don't know the value of infinity, we couldn't define { 1 }^{ \infty }.

{ 1 }^{ \infty } \rightarrow Ind

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.