let's talk about the greatest integer function also known as the floor function and it looks like this this is the greatest integer of x now how can we apply it so for example what is the greatest integer of 0.7 how can we evaluate that so think of all the numbers that are less than 0.7 which of these numbers is the greatest the greatest integer that is less than 0.7 is 0. a technique that can help you figure this out is to draw a number line let's say this is zero one two three four 0.7 is right here on the number line so we need to do is pick the integer that is to the left of it all of these integers are less than 0.7 but the greatest of these is 0. so the greatest integer of 0.7 is 0. now what about the greatest integer of 1.8 what do you think that's going to be well this answer has to be one if we plot 1.8 on the number line all you need to do is pick the integer to the left and that will give you 1. what about the greatest integer of 2.3 this will equal 2. 2.3 is here if you go to the left the next integer is 2. but now let's move on to some negative values let's try a negative 1.6 what is the greatest integer of negative 1.6 and also do this one too negative 2.8 so if we plot negative 1.6 on the number line negative 1.6 is between negative 1 and negative 2. so you need to pick the next integer to the left which is negative 2. negative 1.6 is not greater than negative 1 but it is greater than negative 2. now for negative 2.8 if we plot it it's between negative two and negative three so if we pick the next integer to the left this will give us negative three now let's talk about how to graph this function so first let's put the marks on the graph so this is positive 4 and that's 4 as well so we know that the greatest integer of let's say point six is zero if we try let's say 0.3 that's also gonna be zero as well so therefore we should have a horizontal line between zero and one which will have a y value of zero so it looks like this it's going to be a closed circle at 0 but an open circle at 1. and then the pattern will repeat that's how you can graph the greatest integer function so for example let's say if we want to find the greatest integer of 1.4 so what you can do is choose an x value of 1.4 so that's between 1 and 2. so 1.4 should be somewhere over here then look at the y value notice that the y value is 1. so the greatest integer of 1.4 is one now let's say if we want to find the greatest integer function of three point two so three point two is between three and four so it should be somewhere over here and it corresponds to a y value of three so hopefully you see how this works now let's try a negative one let's say negative two point four what's the greatest integer of negative two point four negative 2.4 is between negative two and negative three and this corresponds to a y value of negative three you can also just go to the left which will give you negative 3 as well so now you know how to evaluate it using the the graph of the greatest integer function now what about this one what is the greatest integer of 2. so when x is 2 we don't use this one because that's an open circle we need to look for the closed circle which is here and that's equal to 2. so the greatest integer of negative one we would use this function that's also equal to negative one so if it's an integer itself it's going to equal that integer try these practice problems find the greatest integer function of 3.2 7.8 negative 3.2 and also negative 4.7 so go ahead and work on these for the sake of practice the greatest integer function of 3.2 we got to pick the greatest integer that's less than 3.2 so this is going to be 3. the highest integer that's less than 7.8 is 7. now negative 3.2 let's use the number line for this here's negative 3 this is negative 4 negative 2. so negative 3.2 is here so the greatest integer that's less than negative 3 is negative 4. the greatest integer that's less than negative 4.7 is negative 5. now try these two as well the greatest integer function of four is going to be four and the greatest integer of negative 5 is itself negative 5. now let's evaluate it with limits what is the limit as x approaches 0 of the greatest integer function of x so what do you think we need to do here so we need to check the one-sided limits let's start with the left side so what is the limit as x approaches 0 from the left of the greatest integer function so let's plug in a number left of 0 we can use negative 0.1 what is the greatest integer of negative 0.1 if we use the number line here's zero here's negative one negative point one is between zero and negative one so we gotta pick the number to left this is gonna be negative one so as we approach zero from the left it's going to equal negative one and recall you can always use the graph you know the graph looks like this so at negative one well zero to the left if we approach zero from the left side notice that it's equal to negative one using the graph now what about the limit as x approaches zero from the right so what is the greatest integer function of positive point one well positive point one is greater than zero so this is going to be zero and as we approach zero an x value of zero from the right side notice that the y value is zero it's on the x axis so you can analyze it graphically or you can just plug it in you can plug in numbers because the left-sided and the right-sided limit do not match because they're different the limit does not exist what is the limit as x approaches 2 from the left of the greatest integer function of x plus 3x so 2 to the left we can plug in 1.9 so let's focus on this portion first what is the greatest integer of 1.9 so out of all the integers that are less than point nine which one is the greatest this is going to be one so as x approaches two from the left the greatest integer of x will be one and then for three x we can plug in two so it's going to be 1 plus 6 so the final answer is 7. now the way you may want to show your work on the test you may want to separate the greatest integer function of x and also separate 3x so you may want to write the limit twice so we know that the limit as x approaches 2 from the left of the greatest integer function of x it has to be 1. and for this side you can just use direct substitution that's gonna be three times two and in the end you should get seven but some teachers may want you to write out each step first so you may have to do it that way depending on what teacher you have what is the limit as x approaches negative 3 from the left of 5 minus 2 times the greatest integer function of x so first let's rewrite it as 5 minus 2 times the limit as x approaches negative 3 from the left of the greatest integer function of x now let's focus on this portion of the expression negative 3 from the left what number should we plug in to represent that so here's negative 3 this is negative 4 and this is negative 2. so negative 3 from the left side means that we need to plug in a number that's between negative 3 and negative 4. so we can plug in negative 3.1 so what is the greatest integer of negative 3.1 it's a negative 3.1 which we already plotted the greatest integer is going to be to the left of that so that's going to be negative 4. so since the limit as x approaches negative 3 from the left side of the greatest integer function of x that's equal to negative four we can now find the value of this expression so it's five minus two times negative four negative two times negative four is positive eight and five plus eight is thirteen so 13 is the answer what is the limit as x approaches 3 of this expression 2 minus the greatest integer function of negative x so because we're dealing with three from either side we need to check the left side and the right side so let's start with the left side first so this is going to be 2 minus the limit as x approaches 3 from the left of the greatest integer function of negative x so if we plug in a number that's to the left of 3 that would be like 2.9 but now notice that we have a negative x on the inside so what we need to do is find out the greatest integer of negative 2.9 negative two point nine is greater than negative three but less than negative two so the answer is negative three and you can always use the number line for that so here's negative two this is a negative three and here's negative one negative two point nine is just to the right of negative three so to find the greatest integer always look to the left and so that's negative three so this should equal negative three so we have two minus negative three so on the left side it's equal to positive five now let's evaluate it on the right side so on the right side we need to plug in a number that's greater than 3 because x approaches 3 from the right so greater than 3 is 3.1 but now to plug in 3.1 the sign is going to reverse it's going to become negative 3.1 so now what is the greatest integer of negative 3.1 negative 3.1 is between negative 3 and 4. so negative 4 is less than negative three but it's greater than all of the integers that are less than negative three so this is negative four so this becomes two minus negative four so the right side of limit is going to be two plus four which is six now because these two do not match we could say that the limit does not exist let's try one more problem let's try the limit as x approaches negative two from the left of the greatest integer function of negative x so try this problem pause the video negative two from the left what number should we use to represent it so here's negative 2 negative 3 negative 1. so we need to pick a number that's to the left of negative 2. so negative 2.1 could be a good representation of something that's to the left of negative 2. so let's plug in negative 2.1 so we have a negative and then negative 2.1 so we're looking for the greatest integer of positive 2.1 because the two negative signs will cancel 2.1 is to the right of 2. so the greatest integer is going to be positive 2. so therefore that's the answer the limit as x approaches negative 2 from the left side which is like 2 negative 2.1 and if we plug that into negative x that becomes positive 2.1 which the greatest integer of that is positive 2. so that's the answer for this question you