My problem revolves around the function:
$$ f(x) = \frac{\sin(\tan x) - \tan(\sin x)} {\arcsin(\arctan x) - \arctan(\arcsin x)} $$
The $\lim_{x\to 0}f(x)= 1$. However, whilst approaching 0, there is a lot of sporadic oscillations (mainly between -0.005 and 0.005). I've been asked
'Since the limit exists, how do you explain the outputs from the function for "small" values of x and for the plot that you generated?'
I can't really give a justifiable reason for it other than because the top and bottom functions equal what they equal and that is what they give you, but I have a feeling that isn't what we were going for...
PS here is a graph of it for quick reference:
This is a typical example of the loss of significance occurring in numerical calculations. Let's focus on the numerator. When $x=0.005$, the terms being subtracted, $\sin \tan x$ and $\tan \sin x$, are of size $\sim 0.005$ themselves. In double precision, their representation incurs errors of size $0.005\cdot 2^{-53}\approx 5\cdot 10^{-19}$.
The Taylor series of $\sin \tan(x)-\tan\sin x$ at $x=0$ begins with $-x^7/30$. Plugging $0.005$ in gives a number about $2\cdot 10^{-18}$: this is the magnitude that the numerator has.
So, at this size of $x$, the magnitude of the numerator is comparable to the size of round-off errors. Similar thing happens in the denominator. The result of computation is garbage.