I am looking for some help with the following question. I am really stuck!
I am trying to use python to evaluate
$$\frac{\sqrt{9+x}{-3}}{x}$$
for $x = 10^{-n}$, $n \in \{4,...,13\}$ and interpret the observations. Is there a better way to evaluate this expression?
I am not sure how to proceed with this, maybe multiplying by a conjugate and evaluating would be better?
Obviously if you multiply the numerator and denominator by the conjugate expression$\sqrt{9+x}+3$, you can prove that the limiting value is $1/6$. To wit:
$\frac{\sqrt{9+x}-3}{x}=\frac{9+x-3^2}{x(\sqrt{9+x}+3)}=\frac{1}{\sqrt{9+x}+3}\rightarrow 1/6 \text{ as }x\rightarrow 0$
But I suspect this is a computer exercise, meaning you should just punch the expression into your computer or calculator and see what happens.
If your machine is like most machines (all machines if you keep trying lower powers of ten), you seem to approach $0.16666...$ with no problem ... until, when the absolute value of the exponent gets large enough, there are suddenly random deviations that keep getting worse. That is what you should be looking for, and interpreting as ill conditioning when you subtract numbers close to each other.
If you put in the conjugate factor in the numerator and denominator and try again, you kill the deviations. This is an example of removing the ill-conditioned operation.