# 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! :)

Jun 27, 2017

If we are working in $\mathbb{R}$ 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 $\mathbb{R}$, the domain of $\frac{1}{\sqrt{x}}$ includes no value less that $0$, so there is no definition for the limit as $x$ approached $0$ from the left.

Jun 27, 2017

${\lim}_{x \to {0}^{-}} \left(\frac{1}{\sqrt{x}}\right) = \left(- i\right) \infty$

#### Explanation:

Here is my argument:
${\lim}_{x \to {0}^{-}} \left(\frac{1}{\sqrt{x}}\right) \iff {\lim}_{x \to {0}^{+}} \left(\frac{1}{\sqrt{\left(- x\right)}}\right) = {\lim}_{x \to {0}^{+}} \left(\frac{1}{i \sqrt{x}}\right) = {\lim}_{x \to {0}^{+}} \left({i}^{-} \frac{1}{\sqrt{x}}\right) = {\lim}_{x \to {0}^{+}} \left(\left(- i\right) \cdot \frac{1}{\sqrt{x}}\right) = - i \cdot {\lim}_{x \to {0}^{+}} \left(\frac{1}{\sqrt{x}}\right) = \left(- i\right) \infty$

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.