Limit is Haunting! (1)

Relevant wiki: L'Hopital's Rule - Basic

$\begin{aligned} L & = \lim_{n \to \infty} \frac {n^2}{2^n} & \small \color{#3D99F6} \text{A }\infty/\infty \text{ case, L'Hôpital's rule applies.} \\ & = \lim_{n \to \infty} \frac 2{\ln^2 2 \cdot 2^n} & \small \color{#3D99F6} \text{Differentiating up and down w.r.t. }n \text{ twice} \\ & = \boxed{0} \end{aligned}$

Create a table. (I'm not always the best with other types of solutions, so I used the easier method - not as pretty, but it works)

X | f(X)

1 | 0.5

2 | 1

3 | 1.125

4 | 1

5 | 0.781

10 | 0.098

20 | 0

50 | 0

As seen from the table, as $x \rightarrow \infty$ , $f(x) \rightarrow 0$ .