Since the denominator of the expression at x = 2 is 0, we must look at the one-sided limits in order to find the limit at x = 2 if it exists. Then we have,

lim_x→2⁺  (  x  / |x-2| ) =  lim_x→2⁺  (  2  / 0⁺ ) = +∞

lim_x→2^-  (  x  / |x-2| ) =  lim_x→2^-  (  2  / 0⁺ ) = +∞

Hence, since the one-sided limits show the same behavior at x = 2, the limit exist at x =2, and it is +∞.