\ln\lim_{x\to\infty}\left(1-\frac{1}{\frac{n+4}{3}}\right)^n=\ln\lim_{x\to\infty}\left(1-\frac{1}{\frac{n+4}{3}}\right)^{-\frac{n+4}{3}\cdot{(-\frac{3}{n+4})}\cdot{n}}= \ln\lim_{x\to\infty}\left[\left(1-\frac{1}{\frac{n+4}{3}}\right)^{-\frac{n+4}{3}}\right]^{\lim_{x\to\infty}\frac{3n}{n+4}}=-\lim_{x\to\infty}\frac{3n}{n+4}=-3