algebra precalculus - Why is $frac{1}{frac{1}{0}}$ undefined?

Is the fraction

$$\frac{1}{\frac{1}{0}}$$

undefined?

I know that division by zero is usually prohibited, but since dividing a number by a fraction yields the same result as multiplying the number by the fraction's reciprocal, you could argue that

$$\frac{1}{\frac{1}{0}} = (1)\left(\frac{0}{1}\right) = 0$$

Is that manipulation permissible in this case? Why or why not?

Answer

Another way to think about this is order of operations:

$$
\frac{1}{\frac{1}{0}}=1/(1/0)
$$

I always compute what's inside the parenthesis first, which gives me undefined, and I have to stop there.