Real Number Sequences and Limits

Abstract

The exponential function “exp (x) = ex” and its corresponding inverse function the natural logarithmic function “ln(x)” are found throughout physics. The exponential function may be defined as the real number “e” to the power of the argument (the real number x). We are then lead to defining the exponentiation operation of a real number base to the power of a real number exponent (in addition to defining the number e). In turn, the exponentiation operation of a real number base to the power of a real number exponent may be defined, and its properties are derived from the exponentiation operation of a real number base to the power of a rational number exponent (discussed in Chap. 5) and from real number sequences and their properties. In this chapter, we will discuss real number sequences and some of their general properties that we will be applying to the definition of the exponentiation operation of a real number to the power of a real number exponent, which, in turn, will lead us to the exponential function.