Answers:
[tex]\displaystyle \lim_{x\to -3^{+}} h(x) = 1\\\\\displaystyle \lim_{x\to -3^{-}} h(x) = 1\\\\\displaystyle \lim_{x\to -3} h(x) = 1\\\\\displaystyle \lim_{x\to 0^{+}} h(x) = 0\\\\\displaystyle \lim_{x\to 0^{-}} h(x) = 3\\\\\displaystyle \lim_{x\to 2^{-}} h(x) = \infty\\\\\displaystyle \lim_{x\to 2^{+}} h(x) = \infty\\\\[/tex]
The limit exists at -3
The limit does not exist at 0
The limit exists at 2, assuming your teacher allows positive infinity to be an answer (otherwise, the limit doesn't exist).
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Explanation:
If we start on the left side of x = -3, and approach toward x = -3 itself, then we will approach y = 1. Imagine it's like a car on a roller coaster able to move along the curve. If the car is to the left of x = -3, then it goes uphill slowly approaching that limiting value.
If we start on the right side of x = -3, and approach -3 itself, then we approach the same y value as before
So that's how I'm getting
[tex]\displaystyle \lim_{x\to -3^{+}} h(x) = 1\\\\\displaystyle \lim_{x\to -3^{-}} h(x) = 1\\\\\displaystyle \lim_{x\to -3} h(x) = 1\\\\[/tex]
The third limit is basically the combination of the first two limits. If the LHL (left hand limit) and the RHL (right hand limit) are equal, then the limit exists.
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We see that the LHL and RHL at x = 0 aren't the same. So the limit does not exist at x = 0
The LHL for x = 0 is 3 while the RHL for x = 0 is 0.
That explains why
[tex]\displaystyle \lim_{x\to 0^{+}} h(x) = 0\\\\\displaystyle \lim_{x\to 0^{-}} h(x) = 3\\\\\\\displaystyle \lim_{x\to 0} h(x) = \text{DNE}\\\\\\[/tex]
DNE means does not exist
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Lastly, when we approach x = 2 from the left, we head upward toward positive infinity.
So [tex]\displaystyle \lim_{x\to 2^{-}} h(x) = \infty\\\\[/tex]
Also, [tex]\displaystyle \lim_{x\to 2^{+}} h(x) = \infty\\\\[/tex] because we're heading upward forever when approaching x = 2 from the right side.
We can then say [tex]\displaystyle \lim_{x\to 2} h(x) = \infty\\\\[/tex]
