Functions and Continuity
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Limits
Let us continue to use the functions g and h defined above. [Equation goes here - download the original pdf to see it.] The intuitive idea behind our arguments about the continuity or discontinuity of these functions involved the notion of a limit. As we get closer and closer to x = 0, the value of [symbol] gets closer and closer to 1, whether we approach x = 0 from the negative end of the x-axis or from the positive end of it. At x = 0 it is unambiguous that [Equation goes here - download the original pdf to see it.]. [Diagram goes here - download the original pdf to see it.] For this argument we write informally [Equation goes here - download the original pdf to see it.] Note here the use of - and + symbols in connection with the symbol for the limit. These indicate from which side of the given x-value we are approaching; the minus (-) indicates we approach it from the negative end of the x-axis and the plus (+) sign indicates that we approach it from the positive end of the x-axis. Because the two limits converge on the same value, the function [Equation goes here - download the original pdf to see it.]is continuous. Before we move on, it is useful to briefly consider what the formal definition of a limit might be, and how the formal argument would go. Limits are primarily defined for sequences of numbers. For example, you may be familiar with the sequence defined by the general term [Equation goes here - download the original pdf to see it.] The sequence is generated by substituting successive values of n [Equation goes here - download the original pdf to see it.] This sequence can be shown to converge on a single value, and is the definition of the important irrational number e. [Equation goes here - download the original pdf to see it.] As n gets larger and larger (approaches infinity) the value of [Equation goes here - download the original pdf to see it.] gets closer and closer to the number [Equation goes here - download the original pdf to see it.] . We can write this also as [Equation goes here - download the original pdf to see it.] This notation does not capture formally the notion of getting "closer and closer". This notion involves the idea that the difference between the value of the sequence and the limit is smaller and smaller. Let us use l to stand for the limit, and the Greek letter (epsilon) to stand for a small number. Let [symbol] stand for the nth term of the sequence. Then we are saying that as n gets larger and large [Equation goes here - download the original pdf to see it.] That is, however, small we make the value of [symbol] is smaller than it, provided that n is large enough. So this is the basis of the formal definition of a limit
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Contents of
Functions and Continuity
1 Functions
2 Graph
3 Inverse of a Function: monotone increasing or decreasing functions
4 The reciprocal function - singularities
5 Functions defined piecewise on their domain
6 Limits
7 Formal definition of a limit
8 Formal definition of the limit of a function at a point x = a
9 Informal arguments
10 Continuity
11 Combining limits
12 Quotients
13 Image set
14 Inverse image
15 Odd and even functions
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