Use the graph of the greatest integer function y = ⌊x⌋, Figure...

Question

Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.

b. limt→4−(t−⌊t⌋)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider the graph of the function F of X is equal to 3X minus the floor function of X to estimate the limit of F of X as X approaches 3 from the left. Awesome. So it appears for for this particular problem, we're trying to figure out what the estimate. We're trying to estimate this limit of F of X as X approaches 3 from the left, using our graph of our function of F of X, which once again is equal to 3X minus the floor function of X. So with that said, now that we know that we're ultimately trying to estimate the limit of F of X as it approaches X approaches 3 from the left, let's quickly look at our multiple choice answers to see what our final answer might be. A is 6, B is 5, C is 7, and D is does not exist or DNE for short. So with that said, Looking at our graph that is provided to us for our function of FF X, we can see that we have a unique curve. We have a curve that has 123, and 4 discontinuities, and these discontinuities, or you might have heard them called as holes, are represented by these white dots with a black border around them. And as you can see, it's technically one curve, but since we have these discontinuities, the curve disappears into the discontinuity or into the hole, and it starts up again at another point on the graph and it forms almost like a stair shape because it's slow it goes up. Then it disappears and then it goes up some more and then it disappears. So it's almost like a jagged line, almost like a lightning bolt or resembles a lightning bolt, almost in a sense, or a very awkward staircase. So now that we have a visualization of what's happening for this particular problem, noting that our curve is represented in red and we have solid points by that are represented by filled in red dots and then holes represented by the white dots with a black ring around them or a black border. We can see that when we when we investigate and examine our graph, we can see that as X approaches 3 from the left, we can see that the Y value will approach 7. So that will mean, without a doubt, that we can go ahead and write that the limit as X approaches 3 from the left, that's what that negative symbol in the power means, of F of X. So the limit as X approaches 3 from the left of F of X is going to be equal to 7, and that is our final answer. All right, we did it, and this corresponds with multiple choice answer letter C, which once again is 7. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.

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b. limt→4−(t−⌊t⌋)

Key Concepts

Greatest Integer Function

Limit from the Left

Piecewise Functions

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