Question

`lim_(x->0^-) (1/sqrt(x)) = ?` How do we calculate this?

I'm going to answer the question myself, but I want you people to validate my arguments, and please share if you have any critique or different approaches! :)

Answer 1

If we are working in `RR` the limit does not exist. (I'd say there is no definition for the limit.)

Explanation:

I am used to definitions of one sided limits that presuppose that the domain includes values on the appropriate side.

In `RR`, the domain of `1/sqrtx` includes no value less that `0`, so there is no definition for the limit as `x` approached `0` from the left.

Answer 2

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

Explanation:

Here is my argument:
`lim_(x->0^-) (1/sqrt(x)) iff lim_(x->0^+) (1/sqrt((-x))) = lim_(x->0^+) (1/(isqrt(x))) = lim_(x->0^+) (i^-1/(sqrt(x))) = lim_(x->0^+) ((-i) * 1/(sqrt(x))) = -i * lim_(x->0^+) (1/sqrtx) = (-i)oo`

Do you approve?

My biggest concern is actually with the very first step. I didn't use mathematical laws, I purely used logic, so I'm not sure if it's legal or not.