d d x lim y → 2 1 y - 2 1 x - 1 x + y - 2 =

Given,

$\frac{d}{dx}\left(\underset{y\to 2}{\lim }\frac{1}{y-2}\left(\frac{1}{x}-\frac{1}{x+y-2}\right)\right)$

$=\frac{d}{dx}\left(\underset{y\to 2}{\lim }\frac{1}{y-2}\left(\frac{x+y-2-x}{x\left(x+y-2\right)}\right)\right)$

$=\frac{d}{dx}\left(\underset{y\to 2}{\lim }\frac{1}{y-2}\left(\frac{y-2}{x\left(x+y-2\right)}\right)\right)$

$=\frac{d}{dx}\left(\underset{y\to 2}{\lim }\left(\frac{1}{x\left(x+y-2\right)}\right)\right)$

$=\frac{d}{dx}\left(\frac{1}{x\left(x+2-2\right)}\right)$

$=\frac{d}{dx}\left(\frac{1}{x^2}\right)$

$=\frac{-2}{x^3}$