Answer: Choice D
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Explanation:
Let's go through the answer choices one by one to see which are true, and which are false.
- Choice A) This is true because as we approach x = 2 from the left hand side, the y values get closer to y = 1 from the top
- Choice B) This is true. As we get closer to x = 4 on the left side, the blue curve is heading downward forever toward negative infinity. So this is what y is approaching when x approaches 4 from the left side.
- Choice C) This is true also. The function is continuous at x = -3 due to no gaps or holes at this location, so that means its limit here is equal to the function value.
- Choice D) This is false. The limit does exist and we find it by approaching x = -4 from either side, and we'll find that the y values are approaching y = -2. In contrast, the limit at x = 2 does not exist because we approach two different y values when we approach x = 2 from the left and right sides (approach x = 2 from the left and you get closer to y = 1; approach x = 2 from the right and you get closer to y = -2). So again, the limit does exist at x = -4; however, the function is not continuous here because its limiting value differs from its function value.
- Choice E) This is true because the function curve approaches the same y value from either side of x = 6. Therefore, the limit at x = 6 exists.
