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Let \( \alpha1 \) and \( \beta-1 \), then the value of
\( \mathrm{P}^{22: 5} \) \( \lim _{n \rightarrow \infty} n^{\beta-\alpha}\left(\frac{1^{\alpha}+2^{\alpha}+\ldots+n^{\alpha}}{1^{\beta}+2^{\beta}+\ldots+n^{\beta}}\right) \) is :

W
(1) \( \frac{\beta+1}{\alpha+1} \)
(2) \( \frac{\alpha+1}{\beta+1} \)
(3) \( \frac{\alpha+2}{\beta+2} \)
(4) \( \frac{\beta+2}{\alpha+2} \)
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