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Growth Rates - Relative Rates of Growth Many of the common...

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Relative Rates of Growth Many of the common functions tend to take bigger and bigger values when the argument is getting bigger. In other words, for many com- mon functions f ( x ) we have lim x + f ( x ) = + . Sometimes this limit condition is all we need. But often a more detailed analysis of the behavior of f ( x ) as x + is desired. In this section we will discuss how to compare the rates at which different functions of x grow as x becomes large. We will consider only the functions whose values eventually become positive as x + . To illustrate this, consider the following two functions. f ( x ) = x 10 g ( x ) = 10 x Both of these functions are increasing to + as x + . However g ( x ) increases much faster, for big x . Indeed, for x = 100 we have f (100) = 100 10 = 10 20 = 100000000000000000000 (21 digits) , while g (100) = 10 100 = 10000 ... 000 (101 digits) . The comparison is even more striking when x = 1000. f (1000) = 1000 10 = 10 30 = 1000000000000000000000000000000 (31 digits) , while g (1000) = 10 1000 = 10000 ... 000 (1001 digits)! Clearly, g ( x ) grows much faster than f ( x ) as x + . The comparison of these functions can be made precise if we use the following definition. Definition 1. Let f ( x ) and g ( x ) be positive for x sufficiently large. We say that f ( x ) grows faster than g ( x ) as x + , if lim x + f ( x ) g ( x ) = + or, equivalently, if lim x + g ( x ) f ( x ) = 0 .
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