The absolute condition number is defined as,

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

and the relative condition number is defined as,

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 f(x)=\sqrt{x}, I get for the absolute condition number that,

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

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

Now, the relative condition number gives,

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

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

Thanks!