It's well known that the scalar function $e^x$, for $x \in (-\infty,0]$ can be approximated by Chebyshev rational approximation. In practice, one wants to use a partial fraction decomposition form like the following: $$R_{\nu}(x)= \alpha_0 + \sum_{i=1}^{\nu} \frac{\alpha_i}{x-\theta_i}$$

where for $\nu = 14$ we have "good" accuracy and the coefficients are listed in several papers. See e.g. this paper.

Now, it would be nice to find a rational approximation of $e^x$ , for $x>0$, similar to the one above. Honestly, I don't think that such an approximation can exist. The main reason is that for $x \rightarrow \infty$ such an expression is bounded, while it shouldn't.

Does anyone know if there exist a similar approximation, or if it is indeed impossible?