What does $f(x)=\lim_{\ell \to \infty} \ell\sin(\frac{x}{\ell})-\sum_{n=1}^{\ell-1}\frac{n\sin(\frac{x}{n})}{\ell-1}$ converge to, if at all?

Question

I was messing around on Desmos when I came across this function involving a limit:

$$f(x)=\lim_{\ell \to \infty} \ell\sin\left(\frac{x}{\ell}\right)-\sum_{n=1}^{\ell-1}\frac{n\sin\left(\frac{x}{n}\right)}{\ell-1}$$

that I couldn't find the answer to.

I graphed an approximation on Desmos and it seemed to approach a sinusoid with amplitude L. Could anyone help me simplify it further? Thanks in advance.

(I'm in Calculus I so if the answer involves something higher-level than that please state it beforehand.)

Answer 1

From the Stolz-Cesaro Theorem, we have

$$\begin{align} \lim_{\ell\to\infty}\frac{\sum_{n=1}^{\ell-1}n\sin(x/n)}{\ell-1}&=\lim_{\ell\to\infty}\ell \sin(x/\ell)\\\\ &=x \end{align}$$

Therefore, the limit of interest is $0$