Formal Definitions of One-Sided Limits
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Question

Formal Definitions of One-Sided Limits

Greatest integer function Find (a) limx→400+ ⌊x⌋ and (b) limx→400− ⌊x⌋; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx→400 ⌊x⌋? Give reasons for your answer.

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the limit as X approaches 100 from the right of the floor of X and the limit as X approaches 100 from the left of the floor of X, and assess whether the limit as X approaches 100 of the floor of X exists by applying the limit laws. For our answer choices, A says as X approaches 100 from the right, it's the limit is 99, as it approaches 100 from the left, the limit is 100, and that the limit does not exist. B says from the left and the right equals 100 and that the limit does not exist. C says from the right it's 100, from the left it's 99, and the limit does not exist. And D says that the limit is the same either from the right or the left. It exists and is equal to 100. Now, let's first determine if the limit from the right and the left exist, and then we can talk about, well, let's determine what those limits are rather, and then we can talk about if it exists, OK? No, from the right, OK. So let's start here. As X approaches 100 from the right, OK, that means X is going to be between 100 and 101, and that means that the floor of X then is going to be equal to 100. Thus this means that the limit as X approaches 100 from the right of the floor of X will be equal to 100. Next, as X approaches 100 from the left, if we think about where X is going to be, that means it's going to be between 99 and 100, OK? So that means the floor of X in this case is going to be equal to 99. Therefore, this means that the limit as X approaches 100 from the left of the floor of X is equal to 99. Now, now that we've figured out the value of the limit as it approaches from the left and the right, the question is, does the limit exist? Well, in our problem, it told us to use the limit lots. What does that mean? Well, we know that a limit exists, if the left hand limit and the right hand limit approaching the same value are equal, OK? If they are, if they are equal, then that means the limit, the two-sided limit exists. Well, in this case, notice here that the limit as X approaches 100 from the right, which is 100, is not equal to the limit as X approaches 100 from the left, which is 99. OK, and we're talking about the floor of X here. So in this case, because the right hand limit is not equal to the left hand limit, that means the limit does not exist. Thus, our limit from the right is 100, our limit from the left is 99, and the limit does not exist. Therefore, C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

One-Sided Limits

Greatest Integer Function

Limit Definitions and Continuity

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