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.
Limits and differentiation are the beginning of the study of calculus, which is an important and powerful method of computation.
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 4-1. Point P represents the point corresponding to
The behavior of y for given values of x near the point
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
indicated by interval AB in figure 4-1, view A. This interval may be written as
The corresponding interval for y is between and includes
We now take a smaller interval, DE, about x = 4 by using values of
and find the corresponding interval for y to be between
These intervals for x and y are written as
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
This is shown by the chart in figure 4-1, 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 x2, 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
but not necessarily at
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.
and we consider only values where
Simplifying the first term, we have
Finally, combine terms as follows:
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 x2 - 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