Sum of floor and fractional-part terms near zero

For any real number $y$, let $[y]$ be the greatest integer less than or equal to $y$ and let $\{y\}=y-[y]$. For $n=1,2,…$, and for $x\in\mathbb R$, let

$$f_{2n}(x)=\begin{cases}\left[\dfrac{\sin x}{x}\right], & x\ne0,\ 1, & x=0,\end{cases}\quad\text{and}\quad f_{2n-1}(x)=\begin{cases}\left\{\dfrac{\sin x}{x}\right\}, & x\ne0,\ 1, & x=0.\end{cases}$$

Then $\lim_{x\to0}\sum_{k=1}^{100}f_k(x)$ equals __________