 Hello there and welcome to the screencast where we're going to do a couple of examples of using L'Hopital's rule where the limits are infinite. So let's take a look at this limit here first. It's the limit as x goes to infinity, if you can read my handwriting there, of the fraction 5x squared minus x plus 1 over 2x squared plus 100. So we're going to see in other words what value, if any, this fraction approaches as the x gets really, really, really large. The best place to go before we do any computation here is to go to a graph of this rational function here. And we're going to take a look and see what the graph of this function here does as x actually goes to infinity. So let's code over to a geograper screen real quick. Here's our geograper screen. And as you can see, I've graphed the function 5x squared minus x plus 1 over 2x squared plus 100. And if this function has a limit as x approaches infinity, what it means is as x gets larger and larger and larger, the values of the function might wiggle around a little bit, but will eventually approach a single long-term value. And if you look at the graph, you can see that it is actually doing that. It appears that perhaps the long-term end behavior of this graph is maybe five halves as x gets larger and larger and larger. It looks like that's what the outputs of the function are approaching. I say it looks like it, we can't necessarily draw the conclusion that that really is five halves just by looking maybe it's something very close to five halves but not exactly equal to it. That's why we need to do something other than a graph. So we're going to now cut back to the slide here and use L'Hopital's rule to actually calculate this limit. So the thing we might want to do first is just directly evaluate the point I'm approaching. Infinity is not really a number, so you can't plug it in, quote-unquote, but we can't talk about what happens to the top and the bottom of this fraction as x gets really large. The top of this fraction is a quadratic polynomial, second-degree polynomial, with a positive leading term, that five there. So that means as x gets really, really large, so does the polynomial. So does 5x squared minus x plus one. So this top here, I'm actually going to change this up a little bit. The top of the fraction itself goes to infinity as x goes to infinity. Now on the bottom, the same thing happens. That's also a quadratic. 2x squared plus 100 and as x gets really large, this bottom gets really, really, really large. So what we have here is an infinity over infinity situation. That is, remember, not a number. It's not equal to one. It's not equal to infinity. It's an indeterminate form, which again means that you can't tell just from the fact that the limit works out to, quote-unquote, infinity over infinity that the limit exists or doesn't exist or equals anything. In fact, we have a guess from the graph that the limit should exist and equal something around five-half. So I don't want to conclude that the limit fails to exist or equals one at this point. I want to use L'Hopital's rule. So let me clear out a few markings here and let's go to it. So L'Hopital's rule says that if you encounter a limit that works out to infinity over infinity, then we can find the same limit by taking the limit as x approaches infinity of the derivative of the top over the derivative of the bottom. Again, this is not the quotient rule. I'm taking two derivatives here, not one. I'm going to differentiate the top and the derivative of 5x squared minus x plus 1 is 10x minus 1. And now the derivative of the bottom, 2x squared plus 100, would just be 4x. So L'Hopital's rule says that the limit on the top and the limit on the bottom are equal to each other. That's kind of the magic of the rule. So the limit on the bottom looks a lot easier, so let's try to evaluate it. Well, if I try to directly evaluate x going to infinity again, I get another infinity over infinity situation. The 10x minus 1, if x gets larger and larger and larger, so does 10x minus 1. That also gets larger and larger. And the same thing is true for 4x. So what I have here is a situation where I need to use L'Hopital's rule again. So I'm going to invoke L'Hopital's rule one more time and say that the limit in the second row is equal to this limit. The limit as x goes to infinity of, and I'm going to apply L'Hopital's rule and take the derivative of the top. The derivative of 10x minus 1 is 10, and the derivative of the bottom is 4. Now this doesn't have any x's in it, so I know what the limit is going to be automatically. It doesn't really matter that x is approaching infinity at this point. This is a constant, so the limit itself is going to be 10 over 4, which of course is equal to 5 over 2, and that jives with our guess that we got from the graph. So in this case, we found the limit as x goes to infinity by applying L'Hopital's rule twice. Let's move on to another slide where we're going to do another problem that's a little different. We're still letting x go to infinity, and this time we're looking at a function that isn't so algebraically nice. It's the natural logarithm of x, the quantity squared over x. Again, let's get a feel for what the answer to this limit question ought to be by looking at a graph. So the graph here shows log x squared over x, and as you can see, as x gets really, really large, it appears that the value of this function approaches zero. So we suspect that the limit actually does exist and equals zero at this point, but the graph is really more evidence than proof. So let's go back to the calculation slide here and try to calculate. So if I directly let x go to infinity here, you can see that the bottom is going to go to infinity because that's just x itself. One thing that is true about the natural logarithm, and I'll just note this off to the side here, is that if you look at the natural logarithm function just by itself and let x goes to infinity, that limit itself is infinite. And what that means is, if you look at the natural logarithm function, let x get larger and larger and larger, the natural logarithm function itself gets larger and larger and larger and it never stabilizes, it never levels off. So if the natural logarithm function by itself never levels off, then certainly if I square it, that's not going to level off either. So this top is going to go to infinity as well. In other words, I have another infinity over infinity situation. That's an indeterminate form, and so we're going to need to use L'Hopital's rule to calculate the limit. So let's do that. So L'Hopital's rule says that the limit upstairs is the same limit of a different expression that I get by taking the derivative of the top over the derivative of the bottom. So the derivative of the top, that would be natural log of x squared. I need to use the chain rule to get its derivative. That would be 2 times the natural logarithm of x times the derivative of the natural logarithm of x, which is 1 over x. And on the bottom, it's very easy. It's just the function x and its derivative is 1. So let me clean up the limit here just for a moment. This would give me the limit as x goes to infinity of the 1 on the bottom doesn't really matter. This would just become 2 natural log x divided by x. So now let's see if we can evaluate this new limit by direct quote unquote substitution. Well, we're in another infinity over infinity situation here because as x gets larger and larger and larger x, the bottom gets larger and larger. And we just said that natural logarithm of x gets larger too. So we're going to have to use L'Hopital's rule again twice. So here's the second and what will be the final application of L'Hopital's rule in this case. It's the same limit as x goes to infinity. I'll need to take the derivative of the top. That would give me 2 over x and the derivative of the bottom is just 1. So dividing by 1 doesn't do anything. This is just going to become the limit as x goes to infinity of 2 over x. Now this is not an infinity over infinity situation. If I think about x getting really, really large this time, the x is going to get really, really large, but the 2 doesn't. Okay, 2 is a constant. It doesn't change. So let's think about what's happening here. I have 2 a fixed number being divided by a quantity that is getting really, really, really, really large. So if you have a fraction and the numerator of that fraction stays put, but the denominator of the fraction grows, then the entire fraction is going to go to 0. And that reasoning gets us to our answer, and that answer is what we expected by looking at the graph. So here again, we have another situation where we encountered a series actually of infinity over infinity indeterminate forms, and we use L'Hopital's rule to progress through each of them individually. Thanks for watching.