Formalizing the concept of ``approaching''

We now will formalize the definition of approaching, and term the formal concept limit. The idea of the definition is to make an analogy with quality control. We want to guarentee that the out put of a process is within certain limits by guarenteeing that the input is within certain limits.

We will take as given the following:

• I is an interval of real numbers;
• $f:I\rightarrow(-\infty,\infty)$ is a function;
• $a\in I$;
• $L\in (-\infty,\infty)$.

We say that the limit as x approaches a of f(x) is L if for every t > 0 there is a m_t > 0 so that if 0 < |x-a| < m_t and $x\in I$ then |f(x) - L| < t. In this case we write

$$\lim_{x\rightarrow\infty}f(x) = L$$

Without any equations the definition reads: We say that the limit as x approaches a of f(x) is L if for every positive tolerance t there is a margin of error m_t so that if

i:

x is not a,

ii:

x is in the domain of f,

iii:

the magnitude of the difference between x and a is less than the margin of error m_t,

than the magnitude of the difference between f(x) and L is less than the tolerance t.

One can reformulate this in slightly different ways, based on whether or not one wants a to be in the domain of f or not. For example, if one says that the domain of f should be I with a removed, then there is no need to say that $x\neq a$, as this is taken care of by saying x must be in the domain of f. On the other hand, should we be faced with a function f defined on an interval not containing the value a of interest we can always apply this definition if a together with the domain of f forms an interval by defining a new function g with g(a) = 0, g(x) = f(x) and with domain a union the domain of f.

For the purposes of this course, the definition of limit is the one stated here. You will be required to state this definition correctly on the midterm and the final exam.

What is important to realize is that the definition of limit is the synthesis of all the properties we ascribed to ''as x approaches a, f(x) approaches L'' and not the other way around!. It is then necessary to verify that these properties hold for ``limits'' as we have defined them here.

We present here some of these verifications. To make life easy, in every case we shall assume that the function f has as its domain all real numbers.