Are these preferences locally non satiated? U(x1,x2)=(x1-1)/(2-x2)^2

Question

I got this utility function representing certain preferences. Are these preferences locally non satiated? Can somebody please explain me with the exact definition of local non satiation why these preferences are locally non satiated or not?

Furthermore, consider that the choice set X= {(x1,x2) element of R2+ : x1>1, x2<2}

Thank you

Answer 1

Preferences represented by $u:X\rightarrow\mathbb{R}$, where $X$ is a non-empty convex subset of $\mathbb{R}^2_+$, satisfy LNS if

$(\forall (x, y)\in X)(\forall \epsilon > 0)(\exists \ (x',y')\in X)(\sqrt{(x-x')^2+(y-y')^2} <\epsilon \ \wedge u(x',y') > u(x,y)) $

Observe that for any given $y\in [0,2)$, $u(x, y) = \dfrac{x-1}{(2-y)^2}$ is strictly increasing in $x$ over $(1, \infty)$. So it satisfy LNS.