Question

Prove that limit lim┬(n→∞)⁡〖(1+1/n)^n 〗=e

Answer 1

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 !