Question

`{ \qquad\qquad\huge\underline{{\sf Question}}}`


`\textsf{State all Indeterminate forms with one example}`
`\textsf{for each of them}` ~ ​

Answer 1

In Mathematics, we cannot be able to find solutions for some form of Mathematical expressions. Such expressions are called indeterminate forms. In most of the cases, the indeterminate form occurs while taking the ratio of two functions, such that both of the functions approaches zero in the limit. Such cases are called “indeterminate form 0/0”. Similarly, the indeterminate form can be obtained in addition, subtraction, multiplication, exponential operations also.

Explanation:

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.

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.

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.

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.

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.

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.

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

hope it helps mate

have a great day

Answer 2

`\huge\mathcal {♨ANSWER♥}`

The term "Indeterminate" means an unknown value. The indeterminate form is a mathematical expression that means that we can't be able to determine the original value even after the substitution of the limits. There are seven indeterminate forms. They are,

1) Infinity over infinity

`( \infty )/( \infty )`

conditions:

`lim \: f(x) \: =\infty \: \: \: lim \: g(x) \: =\infty \\ x -> c \: \: \: \: \: \: \: \: \: \:\: \: x - > c \: \: \: \: \: \:`

2)Zero over zero

`(0)/(0)`

conditions:

`lim \: f(x) \: = 0 \: \: \: lim \: g(x) \: = 0 \\ x -> c \: \: \: \: \: \: \: \: \: \:\: \: x - > c \: \: \: \: \: \:`

3)Infinity minus infinity

`\infty - \infty`

conditions:

`lim \: f(x) \: = 1 \: \: \: lim \: g(x) \: = \infty \\ x -> c \: \: \: \: \: \: \: \: \: \:\: \: x - > c \: \: \: \: \: \:`

4)Zero times infinity

`0 * \infty`

conditions:

`lim \: f(x) \: =0 \: \: \: lim \: g(x) \: = \infty \\ x -> c \: \: \: \: \: \: \: \: \: \:\: \: x - > c \: \: \: \: \: \:`

5)Zero to the power of zero

`{0}^(0)`

conditions:

`lim \: f(x) \: =0 \: \: \: lim \: g(x) \: =0 \\ x -> c \: \: \: \: \: \: \: \: \: \:\: \: x - > c \: \: \: \: \: \:`

6)Infinity to the power of zero

`{ \infty }^(0)`

conditions:

`lim \: f(x) \: = \infty \: \: \: lim \: g(x) \: = \infty \\ x -> c \: \: \: \: \: \: \: \: \: \:\: \: x - > c \: \: \: \: \: \:`

7)One to the power of infinity

`{1}^( \infty )`

conditions:

`lim \: f(x) \: = \infty \: \: \: lim \: g(x) \: =0 \\ x -> c \: \: \: \: \: \: \: \: \: \:\: \: x - > c \: \: \: \: \: \:`

☆...hope this helps...☆

_♡_mashi_♡_