X less than e^x-1 less than x/(1-x) for nonzero real x less than 1

Theorem

The following formula holds for nonzero $x \in \mathbb{R}$ with $x<1$: $$x < e^x -1 < \dfrac{x}{1-x},$$ where $e^x$ denotes the exponential.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.2.33