Is $\frac{1}{ \frac{1}{x-1}}$ equivalent to $x-1$ when $x=1$?

Question

Is $\frac{1}{ \frac{1}{x-1}}$ equivalent to $x-1$ even when $x=1$?

Answer 1

$$\frac{1}{(\frac{1}{x-1})} = \big(\frac{1}{x-1}\big)^{-1} = \big((x-1)^{-1}\big)^{-1} = (x-1)^{(-1)\cdot(-1)} = (x-1)^1 = x-1,\ \ \forall x\neq1$$

The only thing you need to worry about is when $x=1$, in which case the original function is undefined, and all manipulations there after will also be undefined at $x=1$. For convenience, you may choose to define that point as $0$, but this choice must be explicitly stated.