Determine the condition number for taking the square root of a number $x$. Is this problem well-conditioned or ill-conditioned
I know that conditioning measures how sensitive a problem is to a small change in the initial data. The condition number $k$ of a problem $P : R^m \rightarrow R^n$ is the smallest number such
$$\frac{\left| \hat{x_i}-x_i\right|}{\left|x_i\right|} \leq \epsilon $$for all $1 \leq i \leq m$ then
$$\frac{||P(\hat x)-P(x)||}{||P(x)||} \le k \epsilon + 0(\epsilon)$$ where little $0$ is Landau.
I am not quite sure how to use this for my example.
If we let $\hat {x} = \sqrt x (1+\epsilon)$ then
$$\frac{||P(\hat x)-P(x)||}{||P(x)||} = \frac{||\sqrt x (1+\epsilon)-\sqrt x||}{||\sqrt x||}= \frac{||\sqrt x + \sqrt x\epsilon - \sqrt x||}{\sqrt x} = \sqrt x \frac{||\epsilon||}{\sqrt x} = || \epsilon || $$
I found an answer with no detail saying that $k = \frac{1}{2}$ and that it was well-conditioned but I am not sure how my above calculation would how that.
Looking for some help with this, thanks!