$$lim_{x\rightarrow 0}(1+x)^n ~ 1+nx$$
Proof:
Taylor Expansion at x=0
$$ (1+x)^n\approx (1+x)^n|_{x=0} + \frac{d (1+x)^n}{d x}|_{x=0} \times (x-0) + \frac{d^2 (1+x)^n}{d x^2}|_{x=0}\times (x-0)^2 + O(x^3) $$
$$ \approx 1 + nx + \frac{n(n-1)}{2} x^2 + O(x^3)$$
$$ \approx 1 + nx + O(x^2)$$