$$\left\{ \frac{n}{m} + \frac{m}{n} \ \ n,m \in \mathbb{N} \right\} \ \ 0 \notin \mathbb{N}$$
So for $n=1$ $$\lim_{m \to +\infty} \frac{1}{m} +m = +\infty$$ $$\lim_{m \to 1} \frac{1}{m} +m = 2$$
And for $m=1$ $$\lim_{n \to +\infty} \frac{1}{n} +n = +\infty$$ $$\lim_{n \to 1} \frac{1}{n} +n = 2$$
The question is: is this enough to state that $sup = \infty \ $ and $inf = 2$ ?
We have $$\dfrac{m}n + \dfrac{n}m \geq 2$$ with equality holding for $m=n$. Hence, the infimum is $2$. For the supremum, fixing $n$, letting $m\to \infty$, we see that $$\lim_{m \to \infty} \left(\dfrac{m}n+\dfrac{n}m\right) = \infty$$ Hence, the supremum is $\infty$.