# What is the limit as x approaches 0 of 1/x?

Jun 24, 2018

The limit does not exist.

#### Explanation:

Conventionally, the limit does not exist, since the right and left limits disagree:

${\lim}_{x \to {0}^{+}} \frac{1}{x} = + \infty$

${\lim}_{x \to {0}^{-}} \frac{1}{x} = - \infty$

graph{1/x [-10, 10, -5, 5]}

... and unconventionally?

The description above is probably appropriate for normal uses where we add two objects $+ \infty$ and $- \infty$ to the real line, but that is not the only option.

The Real projective line ${\mathbb{R}}_{\infty}$ adds only one point to $\mathbb{R}$, labelled $\infty$. You can think of ${\mathbb{R}}_{\infty}$ as being the result of folding the real line around into a circle and adding a point where the two "ends" join.

If we consider $f \left(x\right) = \frac{1}{x}$ as a function from $\mathbb{R}$ (or ${\mathbb{R}}_{\infty}$) to ${\mathbb{R}}_{\infty}$, then we can define $\frac{1}{0} = \infty$ which is also the well defined limit.

Considering ${\mathbb{R}}_{\infty}$ (or the analogous Riemann sphere ${\mathbb{C}}_{\infty}$) allows us to think about the behaviour of functions "in the neighbourhood of $\infty$".