I was recently trying to plot the graph of the function $f(x)$. This function is defined as follows:
$ f(x) = \frac{\sin{3x} - 3\sin{x}}{(\pi - x)^3} $
I first plotted the numerator ($A(x) = \sin{3x} - 3\sin{x}$) and the reciprocal of the denominator (B(x) = $\frac{1}{(\pi - x)^3}$) separately. Their graphs were as expected, with an asymptote in the latter: image of the graph.
However, when I later plotted the whole function, it showed highly erratic behavior at $x \to \pi$.
The graph is available here.
It can be seen that the function itself is indeterminate at $x = \pi$, but the other two functions ($A$ and $B$) I plotted seem to be quite well behaved at $x = \pi$. This made me wonder why does their product, $f$ behave in an oscillatory fashion. Is it because of the asymptote caused in $B$? Or is there any other reason?
I'd appreciate any insight into the matter.
Here is a graph of the function.
The limit at $\pi$ is $-4$.
Here is a magnified graph.
It looks like it is doing wild stuff near $\pi$. But look at the scale on the $y$-axis... All that wild stuff is within $0.0002$ of the true limit $-4$.