Show that $\frac{5}{n}+e^{-n}=O(\frac{1}{n})$

Question

Show that $\frac{5}{n}+e^{-n}=O(\frac{1}{n})$

I know that $e^n>n, (n\geq 0)$, with which $e^{-n}<\frac{1}{n}$ and so $\frac{5}{n}+e^{-n}<\frac{5}{n}+\frac{1}{n}=6\frac{1}{n}$, then $\frac{5}{n}+e^{-n}=O(\frac{1}{n})$. Is this argument right? Thank you very much.

Answer 1

$$\frac{\frac5n+e^{-n}}{\frac1n}=\frac{5e^n+n}{e^n}\xrightarrow[n\to\infty]{}5$$