Compute $\lim_{n\rightarrow \infty}\frac{\exp(x_j)}{n+\sum_{k=1}^n \exp(x_k)}$ with $x_k$ uniformly bounded

Consider a sequence $\{x_k\}_{\forall k \in \mathbb{N}}$ with $x_k\in [-\bar{M}, \bar{M}]$ $\forall k$ and take the following function $$\frac{\exp(x_j)}{n+\sum_{k=1}^n \exp(x_k)}$$ for some $n\in \mathbb{N}$ and some $j\leq n$.

Can we say something about $$\lim_{n\rightarrow \infty}\frac{\exp(x_j)}{n+\sum_{k=1}^n \exp(x_k)}$$ and, if yes, could you help me to compute it?

$$0 \le \frac{\exp(x_j)}{n+\sum_{k=1}^n \exp(x_k)} < \frac{\exp(x_j)}{n} \le \frac{\overline{M}}{n} = 0,$$ so $$\lim_{n\rightarrow \infty}\frac{\exp(x_j)}{n+\sum_{k=1}^n \exp(x_k)} = 0$$