Capturing the Slope of e^x

The exponential inequality from Cauchy's mean theorem
            
can be applied  to the intriguing pair of functions
            
to determine their limiting values using ordinary algebra.  As x becomes infinitely large, the two functions become closer and closer together.  They never meet for any finite value of x, but converge to the same number.  φ(x) approaches from below, always increasing, and Φ(x) from above, always decreasing. The limit is the Euler number, e.

This is the 12th problem in Heinrich Dörrie's 100 Great Problems of Elementary Mathematics.  Along the way, the following inequality is established:
            
which he uses as the basis for solving the next problem:

 (#13): Transform the exponential function e^x into a progression in terms of powers of x.
A charming inequality (marked (1) below) appears in the introductory steps.  Begin with
            
Now consider an arbitrary number a, and a second number  A = a + δ.   δ is also arbitrary. We establish only that A > a, and the difference between the two is δ.


The next two steps are a roundabout way of implying the Calculus.  Set u = δ, and then u = −δ. This gives
            
Multiply the first expressions by e^a and the second by e^A.
            
A bit of rearranging gives
            
where the value (e^A − e^a)/δ can now be sandwiched between two inequalities:

There are two points, a and A.  A being the greater of the two,  e^a < e^A, which are the leftmost and rightmost terms.   The term in the middle is the slope of the line connecting these two points: rise/run.  We can also make δ as small as we like.  In fact, we can make it 0.
Division by zero is indeterminate, and so it is undefined.  In this case, we can define it.  If δ is zero, then
e^A = e^a+0 = e^a
In other words, we can shrink the distance between the leftmost and rightmost terms until they are the same. The middle term remains trapped between the two, and so at δ = 0, its value, too, must be e^a.

So what does expression (1) mean?  The value of the function e^x at any point x is identical to the slope of the tangent line at that point, for every value of x.  It says,

"e^x is its own derivative."