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CHAPTER 4 LIMITS AND DIFFERENTIATION LEARNING OBJECTIVES Upon
completion of this chapter, you should be able to do the following: 1.
Define a limit, find the limit of indeterminate forms, and apply limit
formulas. 2.
Define an infinitesimal, determine the sum and product of infinitesimals, and
restate the concept of infinitesimals. 3.
Identify discontinuities in a function. 4.
Relate increments to differentiation, apply the general formula for
differentiation, and find the derivative of a function using the general
formula. INTRODUCTION Limits
and differentiation are the beginning of the study of calculus, which is an important
and powerful method of computation. LIMIT
CONCEPT The
study of the limit concept is very important, for it is the very heart of the
theory and operation of calculus. We will include in this section the
definition of limit, some of the indeterminate forms of limits, and some limit
formulas, along with example problems. DEFINITION
OF LIMIT Before
we start differentiation, we must understand certain concepts. One of these
concepts deals with the limit of a function. Many times you will need to find the value of
the limit of a function. The discussion of limits will begin with an intuitive
point of view. We will work with the equation ^{} which is shown in figure 41. Point P represents the point
corresponding to y= 16 and x=4 The behavior of y for given values of x near the point x=4 is the center of the discussion. For the present we will
exclude point P, which is encircled on the graph. We will start with values lying between and including x=2 and x=6 indicated by interval AB in figure 41, view A. This interval
may be written as
The corresponding interval for y is between and includes y=4 and y=36
We now take a smaller interval, DE, about x = 4 by using
values of x=3 and x=5 and find the corresponding interval for y to be between y=9 and y=25 inclusively. These intervals for x and y are written as
and
As we diminish the interval of x around x = 4 (intervals GH and JK) we find the values of
to be grouped more and more closely around y=16 This is shown by the chart in figure 41, view B. Although we have used only a few intervals of x in the
discussion, you should easily see that we can make the values about y group as
closely as we desire by merely limiting the values assigned to x about Because the foregoing is true, we may now say that the limit
of x^{2}, as x approaches 4, results in the value 16 for y,
and we write
In the general form we may write
Equation (4.1) means that as x approaches a, the
limit of f(x) will approach L, where L is the limit of f(x) as x
approaches a. No statement is made about f(a), for it may or may not
exist, although the limit of f(x), as x approaches a, is defined. We are now ready to define a limit. Let f(x) be defined for all x in the interval near x=a but not necessarily at x=a Then there exists a number, L, such that for every
positive number
(epsilon),
however small,
provided that we may find a positive number
(delta) such that
Then we say L is the limit off (x) as x approaches a, and
we write
This means that for every given number
> 0, we must find a number
such that the difference between f(x) and L
is smaller than the number
whenever
EXAMPLE:
Suppose we are given
= 0.1 and
find a
> 0. SOLUTION: We must find a number
such that for all points except we have the
difference between f(x) and 1 smaller than 0.1. We write ^{
} and
and we consider only values where
Simplifying the first term, we have
Finally, combine terms as follows:
so that
or
Therefore,
= 0.3 and we have fulfilled the definition of
the limit. If the limit of a function exists, then
So we can often evaluate the limit by substitution. For instance, to find the limit of the function x^{2}
 3x + 2 as x approaches 3, we substitute 3 for x in the function. Then
Since x is a variable, it may assume a value as close to 3
as we wish; and the closer we choose the value of x to 3, the closer f(x)
will approach the value of 2. Therefore, 2 is called the limit of f(x) as x
approaches 3, and we write
