Unizor - Function Limit - Steepness and Constant 'e'

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Steepness and Constant e

Another important concept we will be dealing with is a concept of a steepness of a curve that represents the graph of a real function y=f(x).

The concept of steepness of a curve at some point was introduced first in the chapter of Algebra dedicated to exponential functions. We have defined it, approximately, as a ratio of the increment of the Y-coordinate along the curve to an increment of the X-coordinate as we step forward from the original point where we want to measure the steepness.

We have also indicated that the smaller increment of X-coordinate (that is, the closer the next point on a curve lies to the point where we measure the steepness) - the better our approximation would be.

Using the terminology we are now familiar with, the steepness of a curve that represents the graph of a real function y=f(x) or, simply, the steepness of a real function y=f(x) at some point x=r is a limit of the ratio of f(r+d)−f(r) to d as d→0,
that is,
lim(d→0)[f(r+d)−f(r)] /d

When we introduced the exponential functions y=ax in the chapter "Algebra - Exponential Function", we have proven that the steepness of function y=a^x at point x=0 depends on the value of its base a, and, for a=2 the steepness of function y=2^x is less than 1, while for a=3 the steepness of function y=3^x is greater than 1.

Assuming that the steepness is smoothly increasing, as we increase the base from a=2 to a=3, it is reasonably to assume the existence of such base between these two values, for which the steepness at point x=0 is equal exactly to 1.

This value, obviously, is not an integer and, as can be proven, not even rational. Traditionally it is designated by the letter e (probably, in honor of a famous mathematician Euler who extensively researched its properties) and is considered as a fundamental mathematical constant, like π.
The approximate value of this constant is 2.71.

Now we can state that
lim(d→0)[er+d−er] /d = 1

The above can be accepted as a definition of a number e. There are a few others, all equivalent to this one, that is each definition can be proven based on another definition.
Here are a few.
lim(n→∞)(1+1/n)^n = e
lim(d→0)(1+d)^(1/d) = e
limn→∞[n·(n!)^(-1/n)] = e
limn→∞(1/0!+1/1!+...+1/n!) = e
The last one can be formulated using infinite summation sign as
Σ(1/n!) = e (sum for n=0, 1,...to ∞)

In this course we will assume the definition of number e as a base of an exponential function y=e^x that has a steepness of 1 at point x=0.

It means that we consider as given the following statement, constituting the defining property of number e.

For any, however small, positive ε exists δ such that
IF |d| ≤ δ
THEN |(e^d−e^0)/d − 1| ≤ ε
or, considering e^0=1,
|(e^d−1)/d − 1| ≤ ε

For those math purists, the existence and uniqueness of this number e was not rigorously addressed. We just relied on the monotonic smoothness of steepness of an exponential function y=a^x at point x=0 as we increase the base from a=2, when this steepness is less than 1, to a=3, when it is greater than 1.

[theoremus]

Thank you Zor. I recommend that you omit angle, when describing steepness. Just use the ratio of rise to run.

[ZorShekhtman]

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