Okay, so, what I want to do now is a problem that I really like. It spins the border. It is a bridge between Calc 1 and Calc 2, in terms of the knowledge needed to solve it. So what I'm going to do is, I'm going to put it down, and I'll I want you to pause the video and think about this for a while. And these problems, this is just one of a couple of kinds. But these problems are ones that usually professors like. Because they, Calc 1 and Calc 2, the line in the middle is sort of arbitrary. Really, it's just based on time. But it's really just calculus, right? All of it is calculus. So, when you have a problem that just ties it all together like this one does, it tends to reappear on exams, or on other things, but, so this is one that I really like. And I want you to pause the video and try it. I should do that now before I go ahead and do the solution. So, pause video. This problem has also appeared on exams here at Hopkins in the past. So, okay, ready? Here comes the solution. So, it's a limit, right? So, it's a limit, first and foremost. So, it's, when was last time you solved limits? It's usually a Calc 1 skill, how to solve limits. But it has an integral in it. So, now we have this, the accumulation function in it and it's a limit with integrals. A nice combination of all things. And if you plug in, if you just blindly plugin, which is sort of the first idea of the limit, you get 1/0, which is infinity. Sort of an internal form 1/0, and then times 0 because if you plug in any of the integral from 0 to 0. So, this is an indeterminate form, right? So, the rookie move, the mistake to make is say, infinity times 0, that's 0. And, of course, that's not true. This is indeterminate, [INAUDIBLE] getting inclusions. And if you dig deep into your Calc 1 memory, you've seen this before, at least examples of this, and this appeared in L'Hospital's rule when you had indeterminate forms, combinations of them. But they weren't in this product form, right? The L'Hospital's rule only works if you have a quotient. So, one of the things that's difficult about this problem, at least the way it's presented, is that it doesn't immediately trigger what to do. So, you can rewrite this problem in a way that does. So, we have a limit as a goes to 0, and I'm going to put the interval on top, from 0 to a, e to the -x squared dx over a. And now when I plug in 0, or hopefully it goes the same. Now when I plug in 0, I have 0/0. And this form is one that most students recognize as L'Hospital's rule. Now, notice by the way, the mistake is to try to workout this integral e to the -x squared is a famous example of an integral that just doesn't work. You can't use these coefficients of integration to solve this thing. So, we have 0/0. So, we're going to leave the integral. And that's okay. And, what do we do? So, we do the L'Hospital's rule. And we know how to do that. So, and here's the beauty of this combination question. I have to take the derivative at the top, and the derivative at the bottom. Obviously, the derivative at the bottom is the easy one, which is 1. Remember a is our variable here. And the derivative at the top, here's a beautiful one about this one, it uses the fundamental theorem of calculus. How do you take the derivative of the accumulation function? Use the fundamental theorem of calculus. This says, replace the variable in the bound, with the variable being integrated. So, this becomes e to the -a squared. So, you plug that in. Now, I have a new limit as every L'Hospital's problem does. It replaces the original limit with a new one. And you ask yourself, is this an easier limit to workout? Now over 1 is, of course, not that exciting, but you can plug in 0, right? You can plug in 0 to get this 1. And so, when you do that, you get e to the -0 squared, 0 squared, 0, -0, 0. This is better known as e to the 0, which of course is 1. So, just wanted to show you guys this kind of question. Hopefully you paused the video and struggled with it for awhile. If you got it right, pat yourself on the back, nice job. If you got it wrong, look for these things. This is how you'll see fundamental theorem questions. Usually when you do some sort of traditional Calc 1 thing that involves a derivative to this function.