[tex] \Big[\sum\limits_{k=a}^n f(k) \Big]' = \sum\limits_{k=a}^n \Big[f(k)\Big]'[/tex]
[tex] \lim\limits_{x\to 0} \dfrac{\sum\limits_{k=1}^m \arctan(k^2 x)}{\sum\limits_{k=1}^m \ln(1+k^3 x)} \overset{\frac{0}{0}}{=} \lim\limits_{x\to 0} \dfrac{\sum\limits_{k=1}^m \dfrac{k^2}{1+k^4 x^2}}{\sum\limits_{k=1}^m \dfrac{k^3}{1+k^3 x}} = \\ \\ = \dfrac{\sum\limits_{k=1}^m k^2}{\sum\limits_{k=1}^m k^3}}= \dfrac{m(m+1)(2m+1)}{6}\cdot \Big(\dfrac{2}{m(m+1)}\Big)^2 = \\ \\ \\ = \dfrac{2}{3}\dfrac{2m+1}{m(m+1)}[/tex]
