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Question Number 138473 by henderson last updated on 14/Apr/21

hi!

calculate : \underset{x \to \infty}{l i m} {[\frac{l n (x + 1)}{l n (x)}]}^{x l n (x)} .

Answered by mathmax by abdo last updated on 14/Apr/21

let f (x) = {(\frac{\ln (x + 1)}{lnx})}^{xln (x)} \Rightarrow f (x) = e^{xln (x) \ln (\frac{\ln (x + 1)}{lnx})}

and \ln (\frac{\ln (x + 1)}{lnx}) = \ln (\frac{lnx + \ln (1 + \frac{1}{x})}{lnx}) = \ln (1 + \frac{\ln (1 + \frac{1}{x})}{lnx})

\sim \ln (1 + \frac{1}{xlnx}) \sim \frac{1}{xlnx} \Rightarrow xlnxln (\frac{\ln (x + 1)}{\ln}) \sim 1 \Rightarrow \lim_{x \to + \infty} f (x) = e