TITLE: Why is $\frac{1}{\frac{1}{0}}$ undefined?
QUESTION [31 upvotes]: 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?

REPLY [4 votes]: Yeah, this is undefined. When we divide some number with some N.D. number it results in N.D. number. 
I think you are making a mistake by considering $1/0$ a fraction. Its not a fraction because denominator is zero, its a number which is not defined. So, here "fraction's reciprocal" doesn't make any sense.
But taking limit of $1/{(1/x)}$ as $x\to 0$ makes sense.