Indeterminate Forms: Introduction

When we deal with the limits of quantities, very often we have to compare numbers such as:

multiplying two large numbers;
subtracting two large numbers;
multiplying a large number with a small number;
etc...

Though it is easy to check that adding two large numbers is a large number (that is ), it is absolutely not clear what happened if we subtract a large number from another large number. We say that we have an indeterminate form. May be one of the most important indeterminate form is the quotient of two small (or large) numbers. Recall that a number close to 0, will be called a small number, while a number close to will be called a large number. Note that is a very large number which happens to be negative. Many are confused about this point since they believe that is the smallest "number" among the real numbers. Again large and small here is to be understood in terms of quantities while the set of real numbers has a natural order which is not of concerns to us here.

Let us give some indeterminate forms which we will take care of in the next pages:

or
; ;

Remember that the inverse of a small number is a large number while the inverse of a large number is a small number, that is

Remark. The inverse of a small number is a large one, this is true sizewise but we do have to work little harder to find out about the sign of the large number ( is a positive large number while is a negative large number). This is the only time when we have to find out about the 0 whether it is positive or negative so we can say something about it inverse. We will write 0+ to designate a positive small number while 0- will designate a negative small number.

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