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Logs of logs Compare the growth rates of ln x, ln (ln
Chapter 4, Problem 68RE(choose chapter or problem)
Logs of logs Compare the growth rates of ln x, ln (ln x), and ln(ln(ln x)).
Questions & Answers
QUESTION:
Logs of logs Compare the growth rates of ln x, ln (ln x), and ln(ln(ln x)).
ANSWER:Solution Step 1 In this problem we have to compare the growth rate of the functions ln x, ln (ln x) and ln (ln (ln x)) Let f(x) = ln x, g(x) = ln (ln x)), h(x) = ln (ln (ln x)). In order to compare the growth rate we will be using the following condition. “If there are two functions f(x)and g(x)both tending to infinity as x . Then the function f grows faster than gas x if the following condition holds f(x) g(x) lim = or lim = 0 ” x g(x) x f(x) l'Hôpital's Rule: f(x) f(x) Suppose that we have one of the following cases, lim g(x)= 0r lim g(x)= ± xa xa Where a can be any real number, infinity or negative infinity. f(x) f(x) In these cases we have lim g(x)= lim gx) xa xa