As we know, the exponential function is defined on real number by
$e^x = 1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$.
Naturally, one can think of exponential function with series $\{x_n\}$ by
$ e^{\{x_n\}} = 1 + x_1 + \frac{x_2^2}{2!} + \frac{x_3^3}{3!} + \cdots$.
Some condition like boundedness may be necessary to $\{x_n\}$.
Let's add more condition, like $x_n>0$ and all $x_n$ are roughly similar to a positive constant $x$.
The question is this:
if above constant $x \to \infty$ and all $x_n\sim x$ what more conditions are necessary for $x_n$ to be $e^{\{x_n\}} \times e^{\{-x_n\}} \asymp 1$ ?
When $\{x_n\}$ is a constant series, then the definition above matches with ordinary exponential function, and above equation would work.
I hope there exists any keyword or some theory about this problem.