Question 1090027
The function is

{{{y=(x^2-4)/abs(x-2)}}} or {{{y=(x+2)*(x-2)/abs(x-2)}}}

The function is clearly undefined at x=2, since the denominator will be 0.  The limit as x approaches 2 will exist if and only if the limits as you approach 2 both from the left and from the right are the same.

For values of x less than 2, |x-2| = -(x-2), so

{{{(x-2)/abs(x-2) = -1}}}

so the function is equivalent to -(x+2), or

  f(x) = -x-2

So as you approach 2 from the left, the function value approaches

  -2-2 = -4

For values of x greater than 2, |x-2| = (x-2), so

{{{(x-2)/abs(x-2) = 1}}}

so the function is equivalent to

  f(x) = 1(x+2) = x+2

So as you approach 2 from the left, the function value approaches

  2+2 = 4

So the answers are

  a) the limit as x approaches 2 from the left is -4
  b) the limit as x approaches 2 from the right is 4
  c) the limit as x approaches 2 does not exist, because the limits from
       the left and right are not the same