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FUNCTIONS, LIMITS, AND CONTINUITY in Visual Studio .NET
FUNCTIONS, LIMITS, AND CONTINUITY Denso QR Bar Code Recognizer In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. QR Code JIS X 0510 Drawer In .NET Framework Using Barcode drawer for VS .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications. [CHAP. 3
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where x and y are called the increments in x and y, respectively. ms where h x x0 x. See Fig. 42. y B y
Also this slope may be written
f x0 h f x0 h
Q P _ S f (x0 + h) f (x0) N R
P y = f (x) f (x0) h = Dx
M x0 x0 + h b x x0 x
Fig. 42 Fig. 43 We can imagine a sequence of lines formed as h ! 0. It is the limiting line of this sequence that is the natural one to be the tangent line to the graph at P0 . To make this mode of reasoning precise, the limit (when it exists), is formed as follows: f 0 x lim f x0 h f x0 h!0 h As indicated, this limit is given the name f 0 x0 . It is called the derivative of the function f at its domain value x0 . If this limit can be formed at each point of a subdomain of the domain of f , then f is said to be di erentiable on that subdomain and a new function f 0 has been constructed. This limit concept was not understood until the middle of the nineteenth century. A simple example illustrates the conceptual problem that faced mathematicians from 1700 until that time. Let the graph of f be the parabola y x2 , then a little algebraic manipulation yields ms 2x0 h h2 2x0 h h Newton, Leibniz, and their contemporaries simply let h 0 and said that 2x0 was the slope of the tangent line at P0 . However, this raises the ghost of a 0 form in the middle term. True understanding of 0 the calculus is in the comprehension of how the introduction of something new (the derivative, i.e., the limit of a di erence quotient) resolves this dilemma. Note 1: The creation of new functions from di erence quotients is not limited to f 0 . If, starting with f 0 , the limit of the di erence quotient exists, then f 00 may be constructed and so on and so on. Note 2: Since the continuity of a function is such a strong property, one might think that di erentiability followed. This is not necessarily true, as is illustrated in Fig. 43. The following theorem puts the matter in proper perspective: Theorem: If f is di erentiable at a domain value, then it is continuous at that value.

