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- [Instructor] What we're gonna 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 I'll build our intuition for even what a limit actually is. So we want to know the limit as x approaches two of x to the third minus 2x squared over 3x minus six. Now, the first thing you might wanna check out is, well, what is this expression equal to when x is exactly equal to two? And we could do that by substituting x with two, so it'll be two to the third power minus two times two squared over three times two minus six. Well, this numerator over here, this is eight minus two times four minus eight, so that's going to be equal to zero, and then the denominator over here is six minus six. Well, that's a zero, and we end up in the indeterminate form right over here. So this expression is not defined for x equals two, but we can think about what does the expression approach as x approaches two. And first, let's think about it visually. So, if we were to graph it, and I graph this on the site 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... Let me do that in a darker color. This is equal to y is equal to x to the third minus 2x squared over 3x minus six. 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 two, and that's why we have this little gap over here, showing that it is not defined. And so what I wanna do is I wanna approximate, well, as x gets closer and closer to two, either from lower values of two or from larger values of two, what is the value of this expression or the value of this function 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 zero, this is two. This is one, 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 zoom in further and further on this graph, so I encourage you to try that out yourself or do it with other graphs. So let's zoom in even more. So once again, we're not defined at x equals two, but here we get a slightly better read. Let's see, this is one, this is two. The value that we are approaching as x gets closer and closer to two, we're getting, our value is 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 one, two, three, four, five. 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 1.3 right over there. So it's a little bit more than 1.3. So a little more than 1.3. So, approximately, 1.3... Now, I do that in a lighter color. So it's approximately 1.3 something, it looks like. Let's zoom in even more to see if we can get an even better approximation. So now, once again, we're approaching that same value. We're not defined at x equals two, but as we're approaching x equals two, let's see. Let me get a darker color. So, it would be right around there, and this is, let's see, this is one, 1.1, 1.2, 1.3, 1.4, 1.5. So once again, it looks like it's about one point... If I were to just base it off of this, it looks like it's about 1.333 or 1.34, so I'd say approximately 1.33. So, approximately 1.33. If I were, it looks like it might be approaching 1 1/3, but we don't know for sure. 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 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 what the graphs can do for us. They got us to about 1.33, 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. So, on this column, I'll have my x, and on this column, I'm going to say, well, what is the expression x minus 2x squared over 3x minute six equals? And we know that when x is exactly equal to two, that this thing right over here isn't defined, but let's see what happens as we approach two. So let's see what happens when we're at 1.9, or 1.99... I'll do 1.99... I'll do 1.999. And we could also see from the other direction. We could say, well, what happens when we approach at 2.001, and see if we are getting... If this, if both of these values seem to be approaching something, and so this is approaching it from lower values of x and then we could say this was approaching it from higher values of x. We could say 2.1, 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 1.9, it's going to be 1.9 to the third power minus two times... Two times 1.9 squared. 1.9 squared is equal to, so we get that value, and then we're going to divide by, divide by, we get, we're gonna divide by three times 1.9 minus six, and that's going to be equal to, looks like it's about 1.203, so I'll just write approximately 1.203. Now, let's try it with a much closer value of x. So that was just 1.9. Now let's go to 1.999. So once again, we're gonna have 1.999 to the third power, and then minus two times, open parentheses, 1.999 squared, close the parentheses, is equal to, so that's my numerator, and then my denominator, so divided by, divided by, and now, open parentheses, three times 1.999 minus six, close my parentheses, is equal to 1.332. So this is interesting. So approximately 1.332. So numerically, I seem to be approaching that same value or close to 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 could even try 1.999999 to see what it actually is approaching. So, for example, if I wanted to try 2.1, so that would be 2.1 to the third power, minus two times, open parentheses, 2.1 squared, and then close parentheses is equal to, that's my numerator, divided by, open parentheses, three times 2.1 minus 6, close parentheses. That's my denominator, and I get 1.47, so approximately 1.47. So now you can think I'm getting closer and closer to two from values larger than two. Let's see if we seem to be approaching the same value. So now let's get even closer to two, 2.01. And I'll do this one, and then I'll leave it up to you to see if you can get even more precise. So, two point... Actually, let's get super precise. Let's, well, let's do three zeroes here. 2.0001 to the third power minus two times, open parentheses, 2.0001. So we're getting really close. Squared, close parentheses, is equal to, so that's my numerator, divided by, open parentheses, three times 2.0001, minus six, this is my denominator. Is equal to, and notice, we're getting closer and closer and closer to 1 1/3, 1.333... Now, five. So, we'll say approximately 1.333, and this is now for, this is actually for 2.0001. So it does indeed look like we are approaching 1.333333, or close to, it looks like we are approaching 1 1/3. But once again, these are just approximations, both through the table or graphically. If you wanna find the exact value of the limit, there are other techniques. We're going to explore those techniques in future videos.