Question

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

Answer 1

The limit does not exist.

Explanation:

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

`lim_(x->0^+) 1/x = +oo`

`lim_(x->0^-) 1/x = -oo`

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

... and unconventionally?

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

The Real projective line `RR_oo` adds only one point to `RR`, labelled `oo`. You can think of `RR_oo` 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(x) = 1/x` as a function from `RR` (or `RR_oo`) to `RR_oo`, then we can define `1/0 = oo` which is also the well defined limit.

Considering `RR_oo` (or the analogous Riemann sphere `CC_oo`) allows us to think about the behaviour of functions "in the neighbourhood of `oo`".