 Hi, I'm Zor. Welcome to Unizor Education. Let's talk a little bit more deeper, I would say, about function limits. Our purpose in this lecture right now is to introduce a concept which is a fundamental concept of calculus. I will not call it derivative, but it will be very much related to derivative. I will use the graphical symbolism and the graphical approach to this, I call it steepness of the curve. And also I would like to introduce the very important constant in calculus, constant E. This lecture is obviously the part of the course of advanced mathematics for teenagers and high school students. It's presented on Unizor.com. I suggest you to watch this lecture from this website and read the detailed notes for this lecture before or after this lecture. So, first of all, let's talk about steepness. Now, the concept of steepness actually was introduced before in this course. When I was talking about function a to the power of x, the exponential function, and in particular we were talking about, okay, this is one, and this is the graph of the function. Now, what the interesting is, and I did discuss this, different a obviously result in different graphs. Now, the greater value of base a leads to a steeper graph at the point zero. So, the steepness is actually measured by the tangential line at this particular point. So, let me just leave only one graph rather than two and we will draw a tangential line. So, if this is the graph, my tangential line goes somewhere like this, right? Now, that was actually a part of the algebra course when I introduced exponential functions. What's very important in this particular case, I was talking about function two to the power of x and function three to the power of x. And it was proven in the algebra part that this function has this particular tangential line at less than 45 degrees angle. And this one greater than 45 degrees angle. Or in other words, the steepness, which is basically measured as the following, you take this point where you want to measure the steepness, you increment argument. So, this is zero, this is d, d is increment. And you compare increment of the function, which is a to the power of x plus d minus a to the power of x divided by increment of the argument, which is x plus d minus x, right? So, this is x, but in this case, x is zero. So, we can actually put just like this. And a to the power of zero is one. So, that would be my steepness with this particular increment of the argument. Now, as we move this particular argument closer and closer to zero, my chord would be closer and closer to the tangential line, right? Because these two points are moving together, they're closing the gap. And my chord will be, if this point moves to this, then it will be this, this, this. And eventually, at some limit, it will be a tangential line. So, using, I did not use the term limit at the time, but now we can. So, now we know that this steepness can be measured as a limit of increment of the function. Well, let me just talk about general function, necessarily. So, limit of this minus this. So, that's increment of the function. So, if this is x, this is x plus d. So, we take increment of the function, which is this one, and divide by increment of the argument. And limit as d tends to zero. This is actually the steepness as we have geometrically kind of implied, expressed in more rigorous terminology of limits. So, again, back to our exponential function, we are talking about this. Minus a to the power of zero is one, which is basically the steepness, steepness at x equals zero. And that's what we are actually interested right now, steepness in this particular point. Okay, now, we have basically proven back in the algebra course that the steepness of this curve, whenever it's less than 45 degrees, is less than one. So, two to the power x has less than one steepness. Three to the power x has greater than one steepness. So, this tangential line is less than 45. If I will put three to the power of x, it would be greater than 45. Now, I will make not really very rigorous assumption that as I'm increasing a from two to three, my tangential line somewhere will be at 45 degrees, and the steepness will be equal to one. It's kind of an assumption actually. So, it's not 100% rigorous. However, it's a very reasonable assumption. You obviously understand that the greater the a, it's smoothly increasing, and it's smoothly increasing the tangential line as well. So, there is some value between two and three where this tangential line is at 45 degrees exactly, and the steepness is equal to one. That particular base of exponential function where the steepness at point x equal to zero equals to one is exactly the number e we are talking about. Now, it's called e most likely in honor of a very famous mathematician Euler or Euler. He was Swiss mathematician. He lived almost his entire life in Russia in St. Petersburg. Basically, one of the founders of Russian mathematical school. So, anyway, I think it was in his honor this constant was called e, and approximately it's equal to 2.71 and then etc. Now, obviously, it's not integer because it's between two and three. It's actually irrational. It can be proven, but that's another story. Anyway, what is important is that we kind of assumed that it exists this particular in the middle point where the steepness is exactly equal to one, and the base, which corresponds to this particular graph, is our number which we want to define called e. It's a very, very fundamental constant in calculus. Lots of different things are related to this. So, anyway, what can we say as a result of this? Well, as a result, we have introduced the number e, irrational number e, and what's the property of this number? We basically have defined it using this characteristic property. So, what's the characteristic property? That e to the power of d minus 1 divided by d tends to 1 as d tends to 0. Okay, so this is a defining property of the number e. And again, I did not really go into theorems of existence and uniqueness of this number. I was just trying to convey certain logical statements which lead to existence of this number e between two and three. Okay, so what's interesting is that the number e can be defined in many other ways because it has many other defining properties, so to speak. And all these definitions are actually equivalent to each other, and one can be derived from another. And let me just put together a couple of other definitions which right now can be considered as properties or as theorems, if you wish. If we assume this is a definition, then other definitions become theorems or vice versa. But I'll just list them without any proof. What's interesting is that this goes to e as n goes to infinity. Now, what else? Well, equivalent to this, if instead of 1 over n we have some kind of a variable d which goes to 0, then n becomes 1 over d. So this is, so here is n goes to infinity, and here d goes to 0. That's another property. Next, next is, next looks kind of complicated. This goes to e as d goes to infinity, n, sorry, n. And one more thing I have here, it's 1 over 0 factorial plus 1 over 1 factorial plus et cetera plus 1 over n factorial plus et cetera. This also goes to e. Or if you wish, sigma 1 over n factorial and from 0 to infinity. That goes to e as well. Sorry, goes to e. So these are very important properties. Each one of them can be proven as a theorem and we might actually derive some of these formulas as well, some of these limits. But in any case, they're all kind of equivalent to whatever I have decided to choose as a definition of the number e. And let me just state at the conclusion of this lecture this definition in epsilon delta language, which is always useful. So what does this mean? It means that for any epsilon greater than 0, there exists such d, that's a d neighborhood of 0 in this particular case, that as long as, well let me just put delta here as usual so we don't get confused, epsilon delta. As long as d minus 0, well basically, so it's just d less than or delta immediately follows from this. So if d is very small in immediate neighborhood of 0, delta neighborhood of 0, that this would be very close to 1, which means absolute value of e d minus 1 divided by d minus 1 would be less than or equal to epsilon. That's what it means. So this is epsilon delta definition, which is basically expressed as a limit here. So for any positive epsilon, there is such a delta, delta neighborhood of 0, if you wish, that as long as our argument is within this neighborhood of 0, small neighborhood of 0, my function would be very close to 1, closer than epsilon. All right, so I consider that the concept of a steepness, which is basically let me just repeat again, steepness of the function at point x is difference between values of function divided by difference between the values of argument. So on the graph, if this is x, this is x plus h, this is f at x plus h, and this is f at x. So we take the difference between values of functions between, so it's this divided by this. So this is basically a tangent of the angle of the chord. But as long as I take the limit of this, if h goes to 0, which means I'm getting closer and closer and closer, then chord actually becomes a tangential line. So I have a tangent of tangential line. Well, tangential line sometimes is called tangent as well, but I don't want to say tangent of tangent being the first tangent being a trigonometric function, and the second tangent being the line. So I'm using tangent and tangential line. So this particular thing is a trigonometric function tangent of a tangential line at point x. Well, obviously, if it exists, if this particular limit exists, because sometimes there is no limit. Let me just give you a very quick example of a function where this does not really exist. Consider the function y equals absolute value of x. It has a graph, this, right? Positive x, it's x. Negative x, it's minus x. So it's always positive. Now, is there a limit of this function increment if x is equal to 0? h goes to 0 in this particular point. Well, not exactly, because if I'm approaching from here, my tangent is actually the same as this one. This would be 45 degrees, and the tangent is equal to 1. If I'm approaching from this, my argument would be negative, but my ordinate would be positive, so it would be negative. So this is 135 degrees. So these are two different tangents if we are approaching 0 from two different sides. Now, in the definition of this limit, I'm not really saying about which side I'm approaching. Maybe from left, maybe from right, maybe from a mixture, like here, here, here, here, here, and then closing, closing. And whenever I'm doing something like this, this function at point x is equal to 0 does not have this limit at all. So it's not always exist. So we're talking about smooth functions, smooths in the terms of this limit exist. Okay, so we have introduced the concept of steepness, which will be obviously converted later on into a concept of derivative of the function. And then we have introduced today a constant e, which is a very, very important constant in calculus. I do suggest you to read the notes for this particular text book, for this particular lecture as a textbook at Unisor.com. That's it. Thank you very much and good luck.