Big theta of function with multiple types of n
You can calculate the limit of the function for $n\to\infty$:
\begin{align}
\lim_{n\to\infty}\frac{n\cdot7^n+\frac{8}{n!}}{(n+7)\cdot7^n}&amp;=\lim_{n\to\infty}\frac{n\cdot7^n}{(n+7)\cdot7^n}+\lim_{n\to\infty}\frac{\frac{8}{n!}}{(n+7)\cdot7^n}\\
&amp;=\lim_{n\to\infty}\frac{n}{n+7}+\lim_{n\to\infty}\frac{8}{(n+7)\cdot7^n\cdot n!}\\
&amp;=1 + 0\\
&amp;=\Theta(1)
\end{align}