2 ## LIMITSA sequence of rational numbers Left-hand and right-hand limits The definition of the limit of a variable The limit of a function of a variable The definition of the limit of a function of a variable Section 2:
vanishing conditions requires the idea of a limit. And central to the idea of a limit is the idea of a sequence of rational numbers. ## A sequence of rational numbersWe encounter such a sequence in geometry when we determine a formula
for the area of a circle. To do that, we inscribe in the circle a regular polygon of 0 That is the idea of a sequence approaching a limit, or a boundary, which in this example is the area of the circle. Problem 1. The student surely can recognize the number that is the limit of this sequence of rational numbers. 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, . . . To see the answer, pass your mouse over the colored area. π We speak of a sequence being infinite, which, in analogy with the sequence of natural numbers, is a brief way of saying that, because of a rule or a pattern or a procedure, there is no limit to the number of terms we coulld name. ## The limit of a variableConsider this sequence of values of a variable 1.9, 1.99, 1.999, 1.9999, 1.99999, . . . Now, no member of that sequence will every equal 2. We say, however, that those values are approaching 2 as their "limit." Why? Because 2 is the In other words, it will be possible to name a term of that sequence such that the absolute value of the difference between that term and 2 -- |1 -- will be less that any positive number we name, however small. (Definition 2.1, below.) (We write the absolute value because the terms are less than 2, and so the difference itself will be negative.) When the values of (How else would We also say that a sequence converges to a limit. The sequence above converges to 2. By a sequence in what follows, we mean an ordering of rational numbers according to a rule or an indicated pattern. Here, for example, is a sequence that approaches 0: 0 ## Left-hand and right-hand limitsNow the sequence we chose were values
But we can easily construct a sequence of values of 2.2, 2.1, 2.01, 2.001, 2.0001, 2.00001, . . . In this case, we write But again, no matter what small number we specify, if we go far enough out in that sequence, the value of a difference | Again, when we say that the values of We summarize this in the following definition. But first,
DEFINITION 2.1. The limit of a variable. We say that a sequence of values of a variable v approaches a limit l (a number not a term in the sequence) if, beginning with a certain term v When that condition is satisfied, we write v l.
And so when the values of a variable approach a limit, there is always a difference between the limit and those values. But that difference can be made as small as we please. That is the essence of a variable approaching a limit. If Δ ## The limit of a function of a variableWe have defined the limit of a variable, but what we often have is a
Now, a sequence of values of "The limit of In fact, let us see what happens to 1.9, 1.99, 1.999, 1.9999, 1.99999, . . .
1.9 It is easy to see that That is, if we go far enough out in the sequence of values of |1.9 -- will become less than any positive number we specify, however small. The definition of the limit of a variable will be satisfied. Moreover, if 2.2, 2.1, 2.01, 2.001, 2.0001, 2.00001, . . . then those values cause 2.2 That sequence also will approach 4. Therefore 4 is the limit of To summarize:
DEFINITION 2.2. The limit of a function of a variable. We say that a function f(x) approaches a limit L as x approaches c if the sequence of values of x, both from the left and from the right, causes the sequence of values of f(x) to satisfy the definition of "approaches a limit L": Definition 2.1. If that is the case, then we write: "The limit of
Thus for the limit of a function to
if and only if
When we say, then, that a function approaches a limit, we The most important limit -- the limit that differential calculus is about -- is called the derivative. All the other limits studied in Calculus I are logical fun and games, never to be heard from again. Now here is an example of a function that does not approach a limit: As Please make a donation to keep TheMathPage online. Copyright © 2016 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |