For the following question, should the approach just be find the limit of $f(x)$ and use that?
Any advice to tackling this?
The Fourier Sine series of the function
$$f(x)= \frac{1}{2}+\frac{|4x-\pi|}{4x-\pi}\cos(x), \;\;\; 0\leq x\leq \pi, $$
converges. Without using a series or integration, find the limit of the Fourier Sine series of $f(x)$ and sketch the limit for $-3\pi\leq x\leq 3\pi$.
The function you have specified is smooth in $[0,\pi]$ except at $x=\pi/4$, and extends periodically to a function with left- and right- limits everywhere, and left- and right- derivatives everywhere. So the Fourier series for the function converges to the mean of the left- and right-hand limits at every point.