Answer
The limit does not exist.
Work Step by Step
Question: Show that $\lim\limits_{(x,y) \to (1,1)}\frac{y-x}{1-y+\ln{x}}$ does not exist.
Check for when $x=1$:
$\lim\limits_{(1,y) \to (1,1)}\frac{y-1}{1-y+\ln{1}}=\lim\limits_{(1,y) \to (1,1)}\frac{y-1}{1-y+0}=\lim\limits_{(1,y) \to (1,1)}\frac{y-1}{-(y-1)}=-1$
Check for when $y=x$:
$\lim\limits_{(x,x) \to (1,1)}\frac{x-x}{1-x+\ln{x}}=\lim\limits_{(x,x) \to (1,1)}\frac{0}{1-x+\ln{x}}=0$
Since the two limits above have different values, we can conclude that $\lim\limits_{(x,y) \to (1,1)}\frac{y-x}{1-y+\ln{x}}$ does not exist.
