Consider x to 2, with x>2 (from the right). Then x-2>0.
So it will be positive infinity.
Consider x to 2, with x<2 (from the left). Then x-2<0.
So it will be negative infinity.
This means the limit doesn't exist, because the limit is not the same from the right and from the left.
It's not good enough to say x - 2 --> 0 etc.
You need to consider and , whether the limiting value of x - 2 etc. is or and consequently whether the left and right handed limits are +oo or -oo ....