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Current time:0:00Total duration:11:04

or any do in this video is see how we can approximate limits graphically and using tables in the future we're also going to be able to learn techniques where we're going to be able to directly figure out exactly what this limit is but for now let's think about how to approximate it and it'll give a little build our intuition for even what is limit actually is so we want to know the limit as X approaches 2 of X to the 3rd minus 2x squared over 3x minus 6 now the first thing you might want to check out is well what is this expression equal to when X is exactly equal to 2 and we could do that by substituting X with 2 so it'll be 2 to the 3rd power minus 2 times 2 squared over 3 times 2 minus 6 well this numerator over here this is 8 minus 2 times 4 minus 8 so that's going to be equal to 0 and then the denominator over here is 6 minus 6 well that's 0 and we end up in the indeterminate form right over here so this expression is not defined for x equals 2 but we can think about what is the expression approach as X approaches 2 and first let's think about it visually so if we were to graph it and I graph this on the sight desmos which has a nice graphing calculator you see the curve y equals this expression right over here so this is the curve of y is equal to let me do that in a darker color this is equal to Y is equal to X to the 3rd minus 2x squared over 3x minus 6 and you can see at least everywhere where I've shown it here and it's actually true it's defined everywhere except for when X is equal to 2 and that's why we have this little gap over here showing that it is not defined and so what I want to do is I want to approximate well as X gets closer and closer to either from lower values of 2 or from larger values of 2 what is the value of this expression or the value of this funk approaching and at this level of zoom it looks like it's approaching this value right over here so as X gets closer and closer to two it looks like our function is getting closer and closer to that value there regardless of which direction we are approaching from and so just approximating it visually right over here let's see this is 0 this is 2 this is 1 right over here this would be 1.5 at this level of zoom it looks like it's about 1.3 or 1.4 so 1.3 or 1.4 let's zoom in a little bit more and if you have access to a graphing calculator or you go to a website like desmos or Wolfram Alpha you can you can zoom in further and further on this graph so I encourage you to try that out yourself or we'll do it with other graphs so let's zoom in even more so once again we're not defined at x equals 2 but here we get a slightly better read let's see this is 1 this is 2 the value that we are approaching as X gets closer and closer to 2 we're getting our values getting closer and closer to that right over there and if we look at what Y value that is let's see if this is this is split into 1 2 3 4 5 so this is 1.2 right over here this is 1.4 right over here so it looks like it's between so this would be 1 point 3 right over there so it's a little bit more than 1.3 so little more than 1.3 so approximately 1.3 let me now do that in a lighter color so it's approximately 1.3 something it looks like let's do them in even more to see if we can get even a better approximation so now once again we're approaching that same value we're not defined at x equals 2 but as we are approaching x equals 2 let's see let me get a darker color so it would be right around there and this is to see this is 1 1.1 1.2 1.3 1.4 1 five so once again it looks like it's about one point if I were to just base it off of this looks like it's about one point three three three or one point three four so I'd say approximately one point three three so approximately one point three three if I were it looks like it might be approaching one and one-third but we don't know for sure where we get remember when you're trying to figure out a limit from a graph the best you can really do here is just approximate try to eyeball well the closer word X gets to two it looks like this function is approaching this value right over here now another technique which tends to be a little bit more precise is to try to approximate this limit numerically so let's do that so let me get rid of these graphs here we already got a sense of of what the graphs can do for us they got us to a about one point three three but now let's try to do it numerically so I'm going to set up a table here and I encourage you to do the same and so on this column I'll have my X and on this column I'm gonna say well what is the expression X minus two x squared over three X minus six equals and we know that when X is exactly equal to two that this thing right over here isn't defined let's see what happens as we approach 2 so let's see at what happens when we're at one point nine or one point nine nine I'll do one point nine nine also one point nine nine nine and we can also see from the other direction we could say well what happens when we approach from at two point zero zero one and see if we are getting if this at both of these values seem to be approaching something and so this is approaching it from from lower values of X and then we could say this is approaching it from higher values of X we could say two point one right over here so let me get a calculator out and evaluate these and I encourage you to do the same get a calculator out and see if you can evaluate these things all right so let's see if we can evaluate it when x equals one point nine it's going to be 1.9 to the third power minus two times two times one point nine squared one point nine squares is equal to so we get that value and then we're going to divide by divided by we get divided by three times one point nine minus six and that's going to be equal to looks like it's about one point two oh three so I'll just write approximately one point two zero three now let's try it with a much closer value of x so that was just one point nine now let's go to one point nine nine nine so once again we're going to have one point nine nine nine to the third power and then minus two times open parenthesis one point nine nine nine squared close the parenthesis is equal to so that's my numerator and then my denominator so / / and now open parenthesis three times one point nine nine nine minus six close my parenthesis is equal to one point three three two so this is interesting so approximately one point three three two so numerically I'm see I seem to be approaching that same value or close that same value that I was approaching graphically and we can also do it from values of x greater than two let me get my calculator back and I'll do some of it and I encourage you to finish this up on your own and you can even try one point nine nine nine nine nine nine to see what it actually is approaching so for example if I wanted to try to point one so that would be two point one to the third power - two times open parenthesis two point one squared and then closed parenthesis is equal to that's my numerator divided by open parenthesis three times two point one minus six close parenthesis that's my denominator and I get one point four seven so approximately one point four seven so now you could think I'm getting closer and closer to from values larger than two let's see if we seem to be approaching the same value so now it's getting even closer to two two point zero one and I'll do this one and then I'll leave it up to you to think to see if you can get even more precise so two point actually let's get super precise let's well let's do let's do three zeros here two point zero zero zero one to the third power minus two times open parenthesis two point zero zero zero one so we're getting really close squared close parenthesis is equal to so that's my numerator divided by open parenthesis three times two point zero zero zero one minus six this is my denominator is equal to notice we're getting closer and closer and closer to one and one third one point three three three three now five so I'll say approximately one point three three three and this is now for this is actually for two point zero zero zero one so it does indeed look like we are approaching one point three three three three three three or close to it looks like we are approaching one and one-third but once again this is these are just approximations both through the table or graphically if you want to find the exact value of the limit there are other techniques we're going to explore those techniques in future videos