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x 0+ in Visual Studio .NET
x 0+ QRCode Recognizer In VS .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Paint QR Code In .NET Framework Using Barcode maker for .NET Control to generate, create QR Code image in .NET framework applications. lim f (x) = +
Denso QR Bar Code Recognizer In .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Barcode Encoder In .NET Using Barcode drawer for Visual Studio .NET Control to generate, create bar code image in VS .NET applications. x 0 Scan Bar Code In .NET Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications. QR Code JIS X 0510 Maker In C# Using Barcode generation for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. lim f (x) = 0 Quick Response Code Creator In .NET Using Barcode drawer for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications. QR Code Printer In Visual Basic .NET Using Barcode drawer for VS .NET Control to generate, create QR Code image in VS .NET applications. Fig. 93 When f (x) has an in nite limit as x approaches a from the right and/or from the left, the graph of the function gets closer and closer to the vertical line x = a as x approaches a. In such a case, the line x = a is called a vertical asymptote of the graph. In Fig. 94, the lines x = a and x = b are vertical asymptotes (approached on one side only). GS1 DataBar Expanded Printer In VS .NET Using Barcode creation for VS .NET Control to generate, create GS1 DataBar Limited image in VS .NET applications. Encode UCC128 In .NET Framework Using Barcode printer for VS .NET Control to generate, create UCC  12 image in VS .NET applications. Fig. 94 Code 128 Code Set C Printer In Visual Studio .NET Using Barcode drawer for Visual Studio .NET Control to generate, create Code 128 Code Set A image in VS .NET applications. Intelligent Mail Generation In VS .NET Using Barcode creation for .NET Control to generate, create 4State Customer Barcode image in .NET framework applications. SPECIAL LIMITS
Drawing Barcode In Java Using Barcode generator for BIRT reports Control to generate, create bar code image in BIRT reports applications. Bar Code Creator In Java Using Barcode encoder for BIRT reports Control to generate, create barcode image in BIRT applications. [CHAP. 9
Data Matrix ECC200 Generator In None Using Barcode drawer for Microsoft Excel Control to generate, create Data Matrix 2d barcode image in Excel applications. UCC128 Maker In Java Using Barcode printer for Eclipse BIRT Control to generate, create GTIN  128 image in BIRT applications. If a function is expressed as a quotient, F(x)/G(x), the existence of a vertical asymptote x = a is usually signaled by the fact that G(a) = 0 [except when F(a) = 0 also holds]. EXAMPLE Let f (x) = Generate Bar Code In Java Using Barcode printer for Eclipse BIRT Control to generate, create bar code image in BIRT applications. Printing DataMatrix In Java Using Barcode encoder for Java Control to generate, create Data Matrix 2d barcode image in Java applications. x 2 for x = 3. Then x = 3 is a vertical asymptote of the graph of f , because x 3
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x 2 = + x 3
x 3 x 2 = x 3
In this case, the asymptote x = 3 is approached from both the right and the left (see Fig. 95). 1 x 2 =1+ . Thus, the graph of f (x) is obtained by shifting the hyperbola y = 1/x three units to [Notice that, by division, x 3 x 3 the right and one unit up.] Fig. 95 LIMITS AT INFINITY: HORIZONTAL ASYMPTOTES
As x gets larger without bound, the value f (x) of a function f may approach a xed real number c. In that case, we shall write x +
lim f (x) = c
In such a case, the graph of f gets closer and closer to the horizontal line y = c as x gets larger and larger. Then the line y = c is called a horizontal asymptote of the graph more exactly, a horizontal asymptote to the right. EXAMPLE Consider the function f (x) = a horizontal asymptote to the right. x 2 whose graph is shown in Fig. 95. Then lim f (x) = 1 and the line y = 1 is x + x 3 If f (x) approaches a xed real number c as x gets smaller without bound,1 we shall write
x lim f (x) = c
1 To say that x gets smaller without bound means that x eventually becomes smaller than any negative number. Of course, in that case, the absolute value x becomes larger without bound. CHAP. 9] SPECIAL LIMITS
In such a case, the graph of f gets closer and closer to the horizontal line y = c as x gets smaller and smaller. Then the line y = c is called a horizontal asymptote to the left. EXAMPLES For the function graphed in Fig. 95, the line y = 1 is a horizontal asymptote both to the left and to the right. For the function graphed in Fig. 92, the line y = 0 (the xaxis) is a horizontal asymptote both to the left and to the right. If a function f becomes larger without bound as its argument x increases without bound, we shall write lim f (x) = + . x +
EXAMPLES
x +
lim (2x + 1) = +
x +
lim x 3 = +
x If a function f becomes larger without bound as its argument x decreases without bound, we shall write lim f (x) = + . EXAMPLES
x lim x 2 = +
x lim x = +
If a function f decreases without bound as its argument x increases without bound, we shall write lim f (x) = . x +
EXAMPLES
x +
lim 2x =
x +
lim (1 x 2 ) = If a function f decreases without bound as its argument x decreases without bound, we shall write lim f (x) = . x EXAMPLES
x lim 2x =
x lim x 3 =
EXAMPLE Consider the function f such that f (x) = x [x] for all x. For each integer n, as x increases from n up to but not including n + 1, the value of f (x) increases from 0 up to but not including 1. Thus, the graph consists of a sequence of line segments, as shown in Fig. 96. Then lim f (x) is unde ned, since the value f (x) neither approaches a xed limit nor does it become larger

