An Alternative Development of Basic Functions

The concept of a limit presents considerable problems to many students, yet often the derivative is defined and l imits taken with l i t t le thought given to the consequences by the instructor, let alone by the student. In this paper we investigate, in a simple problem, the consequence of not proceeding to the limit. Our example indicates how Calculus reduces the range of expressions that occur, yet masks important processes that are of interest and should be returned to later. It reveals a little of the rich area of Mathematics, known as 'basic ' functions, a topic which is so frequently dismissed in a few mutterings about taking the limit. The price that has to be paid for taking the limit occurs when reversing the limit process, then difficulties can arise. 'Basic ' function arise when solving equations involving q-differences, the q-difference of the function f(x) being defined historically as ( ) ( ) () ( ).x1q xfqxfx∆f − So for example where the basic number [n] = 1nn [n]x∆x − =