Show that a Function is Big Theta Using Limits

Question

I'm asked to show that: $f(n) =n^2+ 3n $ is $ \theta$$(n^2)$ using limits.

I know that without limits I can usually solve for a constant, and easily show that this is true, but I'm not too familiar with using limits.

Thus, I'm not exactly sure how to approach this as I'm just learning limits now. Any help would be greatly appreciated.

Answer 1

You want to show $0 < \lim_{n \to \infty} \dfrac{f(n)}{n^{2}} < \infty$. That implies $f(n) \in \Theta(n^{2})$.

Recall:

$$\lim_{n \to \infty} \dfrac{f(n)}{n^{2}} = \lim_{n \to \infty} \dfrac{n^{2}}{n^{2}} + \lim_{n \to \infty} \dfrac{3n}{n^{2}}$$

I assume you can compute the each of these two new limits. Hint- can you cancel terms in the fractions first?