Problem 307 (difficulty: 4/10)

Calculate:
 (1)  \(\displaystyle \lim\root{n^2}\of{1^2+2^n+3^n+\ldots+n^n}=? \)
 (2)  \(\displaystyle \lim\sqrt[n]{\displaystyle 1+\frac12+\frac13+\ldots+\frac1n} = ?\)
 (3)  \(\displaystyle \lim\frac{\displaystyle\frac1{n^2}+\frac1{(n+2)^3}}{ \displaystyle\frac1{n!}-\frac1{\sqrt{(n^2+1)(n^4+2)}}}=?\)

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