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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?

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

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