Does the given set is compact in $\mathbb{R^{2}}$?

Question

Check whether given set on $\mathbb{R^{2}}$ is compact or not ?

$ \left\{(x,y) \in \mathbb{R^{2}}\,\middle|\,x>0,y=\sin\left(\frac{1}{x}\right)\right\}\bigcap\left\{(x,y) \in \mathbb{R^{2}}\,\middle|\,x>0,y=\frac{1}{x}\right\}$

First i have to find the intersection of these two sets $\left\{(x,y) \in \mathbb{R^{2}}\,\middle|\,x>0,y=\sin\left(\frac{1}{x}\right)\right\}$ and $\left\{(x,y) \in \mathbb{R^{2}}\,\middle|\,x>0,y=\frac{1}{x}\right\}$. I know that when value of $x$ increases the value of $\sin\left(\frac{1}{x}\right)\approx\frac{1}{x}$ but question here is that when does $\sin\left(\frac{1}{x}\right)$ will equal to$\frac{1}{x}$. I plot these functions

But at large values graph doesn't zoom in. Do they really intersect or not? If yes then how to find those points?

Answer 1

That set is the empty set (we never have $\sin(x)=x$ when $x\neq0$), and therefore it is compact.