The expressions provided are approximations using Taylor series expansions. Let's break down each approximation:

1. $\ln(1 + 4x)$

Using the Taylor series expansion for $\ln(1 + y)$ around $y = 0$, we get: $\ln(1 + y) \approx y - \frac{y^2}{2} + O(y^3)$

Substituting $y = 4x$: $\ln(1 + 4x) \approx 4x - \frac{(4x)^2}{2} + O((4x)^3)$ $\ln(1 + 4x) \approx 4x - 8x^2 + O(x^3)$

2. $e^{2x} - 1$

Using the Taylor series expansion for $e^y$ around $y = 0$, we get: [ e^y \approx 1 +