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if-lim-n-1-x-7n-29n-2023-find-x-




Question Number 196267 by Rodier97 last updated on 21/Aug/23
          if   lim_(n→+∞)  (1+ (x/(7n)))^(29n) =2023     find x ??
iflimn+(1+x7n)29n=2023findx??
Commented by mr W last updated on 21/Aug/23
find x?  but you said x→+∞
findx?butyousaidx+
Commented by mr W last updated on 21/Aug/23
i guess you meant n→+∞
iguessyoumeantn+
Answered by mr W last updated on 21/Aug/23
 lim_(n→+∞)  (1+ (x/(7n)))^(29n)    =lim_(n→+∞)  [(1+ (x/(7n)))^((7n)/x) ]^((29x)/7)    =lim_(m→+∞)  [(1+ (1/m))^m ]^((29x)/7)   with m=((7n)/x)  =e^((29x)/7) =2023  ⇒x=(7/(29))ln (2023)
limn+(1+x7n)29n=limn+[(1+x7n)7nx]29x7=limm+[(1+1m)m]29x7withm=7nx=e29x7=2023x=729ln(2023)

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