Methods for Finding Asymptotic Lower Bounds
You can codify your method in the following lemma.

Lemma. If $f(n)/g(n) \rightarrow C$, where $C &gt; 0$, then $f(n) = \Theta(g(n))$.

The proof is the same as the one you gave. After you prove this lemma once and for all, you can use it forever. That's actually a good way of verifying $f(n) = \Theta(g(n))$.

Note that the converse to the lemma isn't true. For example, let $f(n) = n$ and let $g(n) = \exp\lfloor\log n\rfloor$. The ratio $f(n)/g(n)$ moves inside the interval $[1,e)$, and in particular does not tend to a constant limit.