Lipschitz Constant of the Smooth Minimum

Question

How does one go about finding the Lipschitz constant of the smooth min function (or demonstrating that it is not Lipshitz). My progress is that we define the smooth minimum as follows $$f(\textbf{x}) = \frac{\sum_{i=1}^n x_i e^{\alpha x_i}}{\sum_{j=1}^n e^{\alpha x_j}}$$ where $\alpha \rightarrow - \infty$. We can easily calculate the partial derivative to be $$\nabla_{x_i} = \frac{e^{\alpha x_i}}{\sum_{j=1}^n e^{\alpha x_j}}\left[1 + \alpha(x_i - f(\textbf{x})) \right]$$. From here, I am unsure how to proceed. Do I simply take the limit as $n \rightarrow -\infty$ of the above? Or can I only demonstrate Lipschitz for a certain range of alpha values? Furthermore do I need to bound the domain in some manner to get Lipschitz such as $\|\textbf{x}\|_1 \leq 1$?