Limited sum

limxlxlxlx2 \Large \lim_{x\to\infty} { \large \dfrac {\displaystyle \sum _{ l\le x }^{ }{ l \left\lfloor \frac { x }{ l } \right\rfloor } }{ { x }^{ 2 } } }

Find the closed form of the above limit to 3 decimal places.


Notation: \lfloor \cdot \rfloor denotes the floor function.


Bonus: Can you find a general formula for limxlxlnxlxn+1\displaystyle \lim _{ x\to \infty}{ \frac {\sum _{ l\le x }^{ }{ {l}^{n} \left\lfloor \frac { x }{ l } \right\rfloor } }{ { x }^{ n+1 } } } for n1? n\ge 1?

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