$\begin{aligned} & \lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}=\lim _{x \rightarrow \infty}\left(1+\frac{5}{x+1}\right)^{x+4} \\ & =e^{5 \lim _{x \rightarrow \infty}\left(\frac{x+4}{x+1}\right)}=x \\ & =e^5\end{aligned}$