Question

I just want to make sure about a concept: What is $\dfrac{0}{0}$ ? Usually anything divided by $0$ is undefined, but a past student of mine reckons they were taught it is $1$ . Is there any theorem or proof for this? Thanks.

Answer 1

Hint: Out of some other indeterminate forms, $\dfrac{0}{0}$ is one of them, which implies it is not defined.
Although we know, any fraction with the same number on the numerator as well as on the denominator is always equal to $1$ , but it’s not true in the case of $0$ .
So, the student may have misunderstood or misunderstood this concept because in any way it cannot be equal to $1$ . Now, we’ll see why it is not defined and cannot be equal to 1.

Complete step-by-step answer:
We are given the indeterminate form$\dfrac{0}{0}$ and we need to show that it is not defined and can be not equal to $1$ . We know, any fraction of the form $\dfrac{x}{x}$ is equal to $1$ .
So, let, if possible, it is equal to $1$ i.e., $\dfrac{0}{0} = 1$ , then using the definition of multiplication, we have that,
$2 \times \dfrac{0}{0} = \dfrac{{2 \times 0}}{0} = \dfrac{0}{0} = 1$ and also, $2 \times \dfrac{0}{0} = 2 \times 1 = 2$ , which implies, $1 = 2$ .
Similarly, we can show $3 = 1$ , $4 = 1$ and so on.
Hence, any number from the number system, say $x$ , can be shown equal to $1$ , which also implies there is no other number except for $1$ , in our number system, which is not possible and this will collapse our number system.
Thus, $\dfrac{0}{0}$ can never be equal to 1 and the form $\dfrac{0}{0}$ is not defined.

Note: Any of the numbers from the number system, divided by zero, is not defined.
Thus, we cannot even say that limits $\dfrac{0}{0}$ can tend to $1$ , or we have to lose the concept of tangent lines and change of rates.
Excluding $\dfrac{0}{0}$ , there are six more indeterminate forms, which are not defined.