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

Theorem

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

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.2.29$