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## iqbal-iqbal 2 years ago Prove that limit lim┬(n→∞)⁡〖(1+1/n)^n 〗=e

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1. luo

Begin with $\lim_{n \rightarrow \infty} \ln(1 + \frac{ 1 }{ n } ) ^{n}$ $= \lim_{n \rightarrow \infty} n . \ln (1 + \frac{ 1 }{ n } )$ $= \lim_{n \rightarrow \infty } \frac{ \ln( 1+ \frac{ 1 }{ n }) }{ \frac{ 1 }{ n } }$ $= \lim_{k \rightarrow 0} \frac{ \ln( 1 + k) }{ k }$(when n goes to infinity ,k = 1/n goes to zero ) $= \lim_{k \rightarrow 0} \frac{ \ln(1 + k) - \ln(1) }{ k }$ ( ln(1) = 0 ) $= ( \ln x) \prime(when,x = 1)$ $= \frac{ 1 }{ x } (when, x = 1)$ $= 1$ $\lim_{n \rightarrow \infty} \ln(1 + \frac{ 1 }{ n } ) ^{n} = 1$ $e ^{\lim_{n \rightarrow \infty} \ln(1 + \frac{ 1 }{ n } ) ^{n}} = e ^{1}$ then get, $\lim_{n \rightarrow \infty} (1 + \frac{ 1 }{ n } ) ^{n} = e$ Proofed !

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