Why does zero raised to the power of negative one equal infinity?

Solution 1:

$$ 0^{-1}=\frac{1}{0}=\mbox{undefined} $$ because $$ \lnot\exists x\in\mathbb{R}:1=0\times x $$

Solution 2:

Strictly speaking we say $0^{-1}$ is undefined.

$x^{-1}$ is the multiplicative inverse of the number $x$.   By definition of the multiplicative inverse, the product of a number and its multiplicative inverse equals one.   So we would have $0\times 0^{-1} = 1$.

However, by definition of zero and multiplication, the product of zero and any number equals zero. So $0\times 0^{-1} = 0$.

So, unless $0=1$, these definitions conflict!   Hence the multiplicative inverse of zero is undefined.

Another way.

We can examine the behaviour of the function $f(x)=x^{-1}$ as $x$ approaches zero.   Plot the curve $y=1/x$ to visualise what is happening.   (It is a hyperbola.)

On the right side, the limit of $x^{-1}$ tends towards positive infinitude as $x$ tends downwards to zero.

$$\lim_{0<x\to 0} \frac 1 x = +\infty$$

On the left side, the limit of $x^{-1}$ tends towards negative infinitude as $x$ tends upwards towards zero.

$$\lim_{0>x\to 0} \frac 1 x = -\infty$$

So there is a discontinuity in the function $f(x)= x^{-1}$ at $x=0$.   Thus the quantity of $0^{-1}$ is indefinite.