For any real number x, the floor function (or greatest integer...

Question

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).

a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

Accepted Answer

Welcome back, everyone. In this problem, we want to find the values for the given limits using the floor function figure. The limit of FFX as X approaches -3 from the left. The limit of FFX as X approaches -3 from the right, the limit of FX as X approaches 2 from the left, and the limit of FX as X approaches 2 from the right. Here we have our floor function figure, Y equals F X. And for our answer choices, A says the limit as X approaches -3 from the left is -2, as X approaches -3 from the right is -1, as X approaches 2 from the left is 4, and as X approaches 2 from the right is 5. B says they are -1, 04, and 5 respectively. C-103 and 4, and D-1-1, 4 and 5 respectively. Now, how are we going to figure out our limits as X approaches the various values from different sides? Well, we can do that by visually inspecting the graph. Here we have a graph of our floor function figure, OK? So let's try to find each of those limits at each of those points. First, let's start with the limit of F of X as X approaches -3 from the left. Now, the limit is the limit of FFX as X approaches the value basically tells us that we want to find FFX value that is being approached as X approaches that specific value. So, in other words, on our graph here, let's go to where X is -3, which is at this point, OK. And now, we're going to look at the value that F of X is approaching as X is approaching -3 from the left, OK? Now, what do you notice? Well, as X approaches -3 from the left, F of X approaches -1. OK. So now we can say that the limit of FX as X approaches -3 from the left equals -1. Now what is the limit as X approaches -3 from the right. Again, we go back to our graph, OK? And notice that as X approaches FFX from the right, OK. Sorry, as X approaches -3 from the right, FFX approaches 0. So no, we also know, or no, we know that the limit of FF X as X approaches -3 from the right equals 0. Next, let's find the limit as X approaches 2 from the left of FF X. We go to the point where X is equal to 2. That here And know, as X approaches 2 from the left, F of X approaches 4. Therefore, that limit equals 4. Finally, the limit as X approaches 2 from the right of F of X. Let's go to our graph and as X approaches 2 from the right, F of X approaches 5. Therefore, the answer here equals 5. Know that we have all our answer choices, if we compare them to our answers, or if we compare all of our results to our answer choices, then we can see that B is the correct answer. Thanks a lot for watching everyone. I hope this video helps.

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).

a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

Key Concepts

Floor Function

Limits from the Left and Right

Discontinuity

Watch next