The numerator is always positive because of the absolute value. What sign is the numerator as x starts off more than 2 and gets smaller and smaller? Well it's 2 minus a number slightly BIGGER than 2, so the sign is negative.
So now we know the sign of the limit is a positive divided by a negative, giving a negative.
As for the value itself, if you were to graph this for x > 2, it would be a horizontal line.... at? -1 like you said.
I don't know exactly what your teacher was looking for, but there's my informal reasoning. Does your teacher want you to use the definition of a limit? I doubt it.