The absolute condition number is defined as,

$\displaystyle Cond_x(f) = \sup_{\delta x} \frac{||f(x+\delta x)-f(x)||}{||\delta x||}$,

and the relative condition number is defined as,

$\displaystyle cond_x(f) = \sup_{\delta x} \frac{||f(x+\delta x)-f(x)||/||f(x)||}{||\delta x||/||x||}$

Now if I look at the problem $\displaystyle f(x)=\sqrt{x}$, I get for the absolute condition number that,

$\displaystyle Cond_x(f) = \frac{1}{2\sqrt{x}}$,

and so,$\displaystyle \lim_{x\rightarrow 0+}Cond_x(f)=+\infty$, and, $\displaystyle \lim_{x\rightarrow +\infty}Cond_x(f)=0$. The way I understand this is that the problem $\displaystyle f$ is well-conditioned for $\displaystyle x>0$.

Now, the relative condition number gives,

$\displaystyle cond_x(f) = \frac{1}{2}$,

and so the problem is well-conditioned for all $\displaystyle x$.. Why is this?

Thanks!