# The number

One of the most important limits in the Calculus is

To prove rigorously that such a limit exists, is beyond the scope of this book. For the present we shall content ourselves by plotting the locus of the equation

and show graphically that, as , the function takes on values in the near neighborhood of , and therefore , approximately.

Figure 3.14: The function .

 -0.1 -0.001 0.001 0.01 0.1 1 5 10 2.868 2.7195 2.7169 2.7048 2.5937 2 1.431 1.0096

As from the left, decreases and approaches as a limit. As from the right, increases and also approaches as a limit.

As , approaches the limit ; and as from the right, increases without limit.

Natural logarithms are those which have the number for base. These logarithms play a very important rôle in mathematics. When the base is not indicated explicitly, the base is always understood in what follows in this book. Thus is written simply or .

Natural logarithms possess the following characteristic property: If in any way whatever,

david joyner 2008-11-22