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Limits of Functions in .NET framework
Limits of Functions Decode USS Code 39 In VS .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET framework applications. Code39 Generation In Visual Studio .NET Using Barcode creator for .NET Control to generate, create Code 3/9 image in .NET applications. some large value of x for which 1/x 2 is smaller than r Therefore, as x grows without bound, 1/x 2 approaches 0, telling us that Lim 1/x 2 = 0 Code 39 Extended Recognizer In .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. Bar Code Creator In .NET Framework Using Barcode creation for .NET framework Control to generate, create bar code image in .NET framework applications. x Bar Code Recognizer In VS .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications. Painting Code 3 Of 9 In C#.NET Using Barcode drawer for .NET framework Control to generate, create USS Code 39 image in .NET applications. Still another example Again, let x be a positive realnumber variable This time, let s evaluate
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It s easy to get mixed up by the meanings of negative direction and positive direction, and how these relate to the notions of left hand and right hand These terms are based on the assumption that we re talking about the horizontal axis in a graph, and that this axis represents the independent variable In most graphs of this type, the value of the independent variable gets more negative as we move to the left, and it gets more positive as we move to the right As we travel along the horizontal axis, we might be in positive territory the whole time; we might be in negative territory the whole time; we might cross over from the negative side to the positive side or vice versa Whenever we come toward a point from the left, we approach from the negative direction, even if that point corresponds to something like x = 567 Whenever we come toward a point from the right, we approach from the positive direction, even if the point is at x = 53,535 The location of the point doesn t matter The important consideration is the direction from which we approach the point Here s a challenge! Consider the base10 logarithm function (symbolized log10) Sketch a graph of the function f (x) = log10 x for values of x from 01 to 10, and for values of f from 1 to 1 Then determine Lim log10 x x 5 Solution
Figure 195 is a graph of the function f (x) = log10 x for values of x from 01 to 10, and for values of f from 1 to 1 The function varies smoothly throughout this span If we start at values of x a little smaller than 5 and work our way toward 5, the value of f approaches log10 5 Therefore, Lim log10 x = log10 5 x 5 Sequences, Series, and Limits
f (x) We close in on this point
from the negative direction
0 5 10 Common logarithm function f (x) = log 10 x
Figure 195 An example of the limit of a function as we approach a point from the negative direction
Memorable Limits of Series
Certain limits of series are found often in calculus and analysis If you plan to go on to Calculus KnowItAll after finishing this book, you re certain to see the three examples that follow! An example Imagine an infinite series where we take a positive integer i and then divide it by the square of another positive integer n Symbolically, we write this as i /n
i =1 When we expand this series out, we write it as 1/n 2 + 2/n 2 + 3/n 2 + + n/n 2 + which simplifies to (1 + 2 + 3 + + n + )/n 2 Memorable Limits of Series
Suppose that we let n grow endlessly larger, increasing the number of terms in the series Let s consider Lim (1 + 2 + 3 + + n)/n 2 As things work out, this limit is equal to 1/2 Therefore Lim
i /n
i =1 = 1/2 Another example Now imagine an infinite series where we square a positive integer i and then divide it by the cube of another positive integer n Symbolically, we write this as

