$\begin{aligned} & \lim _{n \rightarrow \infty} \frac{1^{99}+2^{99}+\ldots .+n^{99}}{n^{100}}=\lim _{n \rightarrow \infty} \sum_{r=1}^n\left(\frac{r^{99}}{n^{100}}\right) \\ = & \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^n\left(\frac{r}{n}\right)^{99}=\int_0^1 x^{99} d x=\left[\frac{x^{100}}{100}\right]_0^1=\frac{1}{100} .\end{aligned}$