Integrability of function

[Galenus]

In lectures I came across a particular statement that got me confused.
I will give full context:

The context is the following proof: given a piecewise continously differentiable L-periodic function $f$, the fourier sereis of f converges to f.
During this proof it must be shown that $$F_x(t) := \frac{f(x+t) - f(x)}{sin(\frac{\pi t}{L})}$$ is Riemann-integrable. Here x is fixed such that f is differentiable at x.

The statement in the proof was merely:

By L'hospital's rule $\lim_{t \rightarrow kL}F_x(t)$ exists, where k is a whole number.

From this (and some hidden steps) it follows that $F_x(t)$ is integrable. But I'm having a hard time finding the missing steps.

I see that L'hospital's rule can be used in this case and that the above limit exists. But I don't understand how that connects to integrability.

My intuition for showing integrability is to define and upper und lower step function and then show that their difference is smaller than some $\epsilon >0$. How else can you show that a function is Riemann-integrable? Maybe in the proof they use the fact that every continious function defined on a bounded interval is integrable? But I don't see how...