Working through my PDE book, it used the following function as an example to introduce piecewise continuity and periodic extensions, and of which to sketch the fourier series:
$$f(x)=\left\{\begin{matrix} 0 & x < \frac{L}{2}\\ 1 & x > \frac{L}{2} \end{matrix}\right.$$
When it shows the computation of the Fourier coefficient $$a_0 = \frac{1}{2L}\int_{-L}^{L}f(x)\,\mathrm{d}x$$ the book uses different limits of integration: $$a_0=\frac{1}{2L}\int_{\frac{L}{2}}^{L}f(x)\,\mathrm{d}x$$
Why do the limits change? Is it for this specific function, or for $a_0$ of any function $f(x)$?
It is for this specific function. Since it is zero for $0<x<L/2$, the integral over that interval vanishes.