Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/46

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22
DIFFERENTIAL CALCULUS

If now approaches the limit zero,must lie between the constant and , which is also .

Therefore , or, . Th. III, p. 18

It is interesting to note the behavior of this function from its graph, the locus of equation

Although the function is not defined for , yet it is not discontinuous when if we define

Case II, p. 15

23. The number . One of the most important limits in the Calculus is

To prove rigorously that such a limit exists, is beyond the scope of this book. For the present we shall content ourselves by plotting the locus of the equationand show graphically that, as , the function

10 1.0096
5 1.4310
2 1.7320
1 2.0000
.5 2.2500 —.5 4.0000
.1 2.5937 —.1 2.8680
.01 2.7048 —.01 2.7320
.001 2.7169 —.001 2.7195

takes on values in the near neighborhood of , and therefore approximately.