For $\alpha \gt -1$ and $\beta \gt -1$, the value of $\lim _{n \rightarrow \infty} n^{\beta-\alpha} \frac{1^\alpha+2^\alpha+\cdots+n^\alpha}{1^\beta+2^\beta+\cdots+n^\beta}$ is

• A

  $\frac{\beta}{\alpha+1}$
• B

  $\frac{\alpha+1}{\beta+1}$
• Correct answer
  C

  $\frac{\beta+1}{\alpha+1}$
• D

  $\frac{\alpha}{\beta+1}$

Hence, the correct answer is option: 'C'.