You are looking for an explanation of WHY choices A & E are true?
Choice A: The graph in the neighborhood of ƒ(4) shows a line coming in from the right to a solid dot at x = 4. This is the very definition of "the limit, as x approaches 4 from the right, of ƒ(x) = ƒ(4).
Choice B: The graph does not have a value at x=0. It has an open dot at (0, 2). This means ƒ(x) is NOT continuous at x=0.
Choice C: The limit, as x approaches 2 from the right, may indeed be 2; however, the limit, as x approaches 2 from the left, is - ∞. Therefore, the limit at x=2 does not exist.
Choice D: Again, the "half-limits," as x approaches -2 from each direction, do not match. Thus, the limit, as x approaches -2 does not exist.
Choice E: The limit, as x approaches 0 from the right is +2. The limit, as x approaches 0 from the left is also +2. The "half-limits" match, so the limit as x approaches 0 is +2, even though the function is not defined at x=0.