Calculus

Using Taylor series expansion: $e^{2x} = 1 + 2x + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \dots = 1 + 2x + 2x^2 + \frac{8x^3}{6} + \dots$ Substituting gives $\lim_{x \to 0} \frac{\frac{4}{3}x^3 + O(x^4)}{x^3} = \frac{4}{3} \approx 1.333$.