TABLE OF CONTENTS How does the limit process apply to the algebraic combination of two or more functions? Sometimes the problems we must confront involve the sum or product of two or more functions, or the quotient of two functions. We will need the following three theorems on limits:
The limit of the sum of two (or more) functions is equal to the sum of their limits:
The limit of the product of two (or more) functions is equal to the product of their limits:
The limit of the quotient of two functions is equal to the quotient of their limits, provided the limit of the denominator is not zero:
These theorems assume that the limits of the two functions exist. However, even though it may be true that neither function separately approaches a limit, the sum, product, or quotient may do so.
The exceptional case of Theorem 3, in which the denominator approaches zero, requires further investigation. Thus, given any quotient g( x)/ f( x) in which f( x) approaches zero, there are two possibilities: g( x) does not approach zero or g( x) also approaches zero.
In the first case, we can make the value of the quotient g( x)/ f( x) greater than any preassigned value by making f( x) sufficiently small, so that the quotient will not approach a limit. In the second case,...
