In "Appendix A: Supporting proofs for approximations" the author justifies the approximation of 1-x with e^(-x) by computing lim_{x→0} (1-x)/e^(-x) = 1.
This is wrong. Consider how this criteria is also satisfied by 1-100000x, since lim_{x→0} (1-x)/(1-100000x) = 1. But this is clearly not a good first-order approximation for 1-x around 0.
The proper justification for replacing 1-x with e^-x around 0 is done by examining the first 2 terms of their Taylor expansions, in other words, the functions' value at 0 and their first derivative at 0. Since these match for 1-x and e^-x, they are good first-order approximations of each other around 0.
This is wrong. Consider how this criteria is also satisfied by 1-100000x, since lim_{x→0} (1-x)/(1-100000x) = 1. But this is clearly not a good first-order approximation for 1-x around 0.
The proper justification for replacing 1-x with e^-x around 0 is done by examining the first 2 terms of their Taylor expansions, in other words, the functions' value at 0 and their first derivative at 0. Since these match for 1-x and e^-x, they are good first-order approximations of each other around 0.