Prove that $$\begin{equation}\lim_{x\to a}\,\sum_{i=1}^N\,f_i(x)=\sum_{i=1}^N\,L_i.\end{equation}$$

Assume that the follow statements are true:
$$\begin{eqnarray} \lim_{x\to a}\,f_1(x)&=&L_1\\ \lim_{x\to a}\,f_2(x)&=&L_2\\ \lim_{x\to a}\,f_3(x)&=&L_3\\ \vdots\\ \lim_{x\to a}\,f_i(x)&=&L_i.\\ \vdots\\ \lim_{x\to a}\,f_N(x)&=&L_N.\\ \end{eqnarray}$$

Then
$$\begin{equation} \lim_{x\to a}\, [ f_1(x)+\cdots +f_i(x)+\cdots +f_N(x) ] = L_1+\cdots +L_i + \cdots +L_N \end{equation}$$
which we can write with our sum ($\sum$) symbol as
$$\begin{equation} \lim_{x\to a}\,\sum_{i=1}^N\,f_i(x) = \sum_{i=1}^N\,L_i. \end{equation}$$