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

Theorem

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

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.2.32