Can We Partition These Limits?

Calculus Level 3

\[ \lim_{n\to \infty} \dfrac{\left(\sqrt1 +\sqrt2 + \cdots +\sqrt n \right) \left( \frac1{\sqrt 1} + \frac1{\sqrt 2} + \cdots + \frac1{\sqrt n} \right)}{1+2+\cdots +n} = \, ? \]

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