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