Question

State true or false: ${0^\infty }$ is an indeterminate form.

Answer 1

Hint: Indeterminate form usually comprises two fractions whose limit cannot be determined by calculating the original limits of the two individual functions. These types of functions are extremely common in calculus. There are many ways by which one can evaluate the indeterminate forms. Some of the methods are the factoring method, L’Hospital’s rule, etc.
There are in total seven expressions that can be included in the indeterminate forms. These are as follows:
$\dfrac{0}{0},\dfrac{\infty }{\infty },{0^\infty },{1^\infty },{\infty ^0},0 \times \infty ,\infty - \infty $ .

Complete step-by-step answer:
It is given that ${0^\infty }$ is an indeterminate form. We need to state if the statement is true or false.
We know that indeterminate forms refer to the fact when limits of two functions cannot be determined solely by the limit of the individual function.
There are only seven indeterminate forms and ${0^\infty }$ do not belong there.
So the reason behind ${0^\infty }$ not being on the list is given below.
We know that if $0$ has anything in its power then the answer will be $0$ itself.
This is because even if we multiply $0$ infinite times the result will give us $0$ .
So, anything in the power of $0$ yields us the result $0$ .
This is not similar to the fact where $0$ is multiplied with infinity.
$0 \times \infty $is indeed an indeterminate form.
Hence, the given statement is False.
${0^\infty }$ is not an indeterminate form.

Note: We should not get confused with the indeterminate form ${\infty ^0}$ . Both the expressions are different and express different meanings. Again, ${1^\infty }$ is an indeterminate form and students should not use it as a basis for interpreting if ${0^\infty }$ is also an indeterminate form.