For $t>0$, let $S_{t}$ denote the ball of radius $t$ centered at the origin in $\mathbb{R}^{n}$. That is,
\[S_{t}=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \sum_{i=1}^{n} x_{i}^{2} \leq t^{2}\right\} .\]
Let $N_{t}$ be the number of points in $S_{t}$ that have integer coordinates, and let $V_{t}$ be the volume of $S_{t}$. Which of the following is $\text{TRUE?}$
- For any $t>0, N_{t}$ is less than the volume of $S_{t}$
- $\lim _{t \rightarrow \infty} \frac{N_{t}}{V_{t}}=\frac{1}{2}$
- $\lim _{t \rightarrow \infty} \frac{N_{t}}{V_{t}}=2$
- $\lim _{t \rightarrow \infty} \frac{N_{t}}{V_{t}}=1$
- $N_{t}$ is a monotonically decreasing function of $t$