Well that is essentially what you're doing, but you can write it out more analytically.

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.