chap analysis techniques how does nfit into thiswe can again take advantage of logarithms obviously <nn so we know that log nis ( log nbut what about lower bound for the factorial functionconsider the following nn ( > ** ** / therefore log >log) / log in other wordslog nis in ohm( log nthuslog nth( log nnote that this does not mean that nth(nn because log log nit follows that log th(log but th( the log function often works as "flattenerwhen dealing with asymptotics (and the antilog works as booster that iswhenever log (nis in (log ( )we know that (nis in ( ( )but knowing that log (nth(log ( )does not necessarily mean that (nth( ( )example what is the growth rate of the fibonacci sequence (nf ( ( for > ( ( in this case it is useful to compare the ratio of (nto ( the following table shows the first few values ( ( )/ ( following this out few more termsit appears to settle to ratio of approximately assuming ( )/ ( really does tend to fixed valuewe can determine what that value must be (nf ( ( - + ( ( ( this comes from knowing that (nf ( ( we divide by ( to make the second term go awayand we also get something