Unizor - Limits - F(x)=e^x

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• published: 12 Dec 2014
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In the chapter describing exponential functions we introduced the concept of steepness β of an exponential function F(x)=a^x at point x=0 as a limit of a ratio between increment of a function between points x=0 and x=1/n (that is F(1/n)−F(0)=a^(1/n)−a^0) and a corresponding increment of an argument (that is 1/n−0) when n→∞, so a point x=1/n is getting closer and closer to a point x=0. After a trivial simplification (since a^0=1) the steepness of a function F(x)=a^x at point x=0 is defined as β = lim[n→∞](a^(1/n)−1)/(1/n) = lim[n→∞]n·(a^(1/n)−1) At this point we would like you to recall the material presented in the lecture Algebra - Exponential Functions - Problems 2 about a steepness of exponential functions 2^x and 3^x at a point x=0. Refer to that lecture for refreshing. We have proved there that the steepness of a function 2^x at point x=0 is less than 1, while the steepness of a function 3^x at point x=0 is greater than 1. Assuming continuity of steepness as it depends on the base of the exponential function, it is reasonable to expect that somewhere between 2 and 3 there is a value of a base of an exponential function with the steepness equaled exactly to 1. This value of a base is denoted as e in mathematics and its approximate value is 2.71. Therefore, we can write: lim[n→∞] (e^(1/n)−1)/(1/n) = 1 Slightly deviating from the rigorousness we used to apply to our statements, we can derive from the last equality that there is an approximation (e^(1/n)−1)/(1/n) ≅ 1 which becomes better and better as n→∞. Or, resolving this for e, e^(1/n)−1 ≅ 1/n or e^(1/n) ≅ 1+1/n or e ≅ (1+1/n)^n Returning to exact equality, we can write the value of e as the following limit: e = lim[n→∞] (1+1/n)^n Summarizing the above logic, we have determined a specific value of the base of the exponential function with the steepness of 1 at point x=0. This value is designated a symbol e and is equal to e = lim[n→∞] (1+1/n)^n Thus, we have defined an exponential function F(x) = e^x Obviously, all properties of exponential functions are preserved in this particular function, including but not limited to: e^0 = 1 e^1 = e e^(x+y) = e^x·e^y e^(x·y) = (e^x)^y Another interesting fact is that steepness, as we introduced it for function F(x)=e^x at point x=0, can be defined at any other point using exactly the same methodology. Consider any point x=x0. Now step forward from this point by a value 1/n to a point x=x0+1/n. Construct a ratio between the difference in the values of a function in these two points and the difference in the values of an argument: β = [F(x0+1/n) − F(x0)] / (1/n) As n→∞, the limit of this ratio characterizes the steepness of a function F(x) at point x0. In case of the function we have introduced in this lecture, F(x)=e^x, the steepness at any point x0 equals to a limit of the following expression, as n→∞: [e^(x0+1/n) − e^x0] / (1/n) = = [e^(x0)·e^(1/n) − e^(x0)] / (1/n) = = e^(x0)·[e^(1/n) − 1] / (1/n) As n→∞, e^(x0) remains a constant while the expression [e^(1/n) − 1] / (1/n), which represents the steepness of function ex at point x=0, has a limit of 1. Therefore, the steepness of function e^x at any point x0 equals to a value of this function at this same point, that is e^(x0). This is quite a remarkable and unique property of function F(x)=e^x.