I dont't understand how the condition number is defined for a problem such as:
$x^2-2xp+1=0,\ p\geq1$ Here there are two roots $x_-=p-\sqrt{p^2-1}$ and $x_+=p+\sqrt{p^2-1}$
I understand that the sum is not well conditionned because the relative error can become very high if the numbers sum up to something close to zero, so when $p$ is close to $1$ both roots $x_-$ and $x_+$ are unstable and when $p$ is big $x_-$ is unstable (that is why we take advantage of the fact that $x_-=\frac{1}{x_+}$ and the division is a stable operation).
But I don't understand why the condition number $K(p)\simeq \frac{p}{\sqrt{p^2-1}}$ for $p>1$
I would appreciate it if someone could explain to me how I should understand the condition number.
By definition, the (relative) condition number of $x_+(p)$ is $$ K(p)=\frac{p}{x_+}\frac{dx_+}{dp} = \frac{p}{p+\sqrt{p^2-1}} \left( 1+\frac{p}{\sqrt{p^2-1}} \right) = \frac{p}{\sqrt{p^2-1}}. $$ The root $x_+$ is sensitive for $p$ close to one with $K$ growing to infinity and insensitive for large $p$ with $K\approx 1$.