lim-1-1-e-

Question Number 53647 by Tawa1 last updated on 24/Jan/19

\lim_{ϕ \to \infty} {(1 + \frac{1}{ϕ})}^{ϕ} = e

Commented by maxmathsup by imad last updated on 24/Jan/19

w e h a v e {(1 + \frac{1}{x})}^{x} = e^{x l n (1 + \frac{1}{x})} b u t l n (1 + \frac{1}{x}) \sim \frac{1}{x} (x \to + \infty) \Rightarrow

x l n (1 + \frac{1}{x}) \sim 1 \Rightarrow {l i m}_{x \to + \infty} {(1 + \frac{1}{x})}^{x} = e .

Commented by Tawa1 last updated on 24/Jan/19

God bless you sir

Answered by tanmay.chaudhury50@gmail.com last updated on 24/Jan/19

t = \frac{1}{ϕ}

\lim_{t \to 0} {(1 + t)}^{\frac{1}{t}} = y (s a y)

\lim_{t \to 0} \frac{l n (1 + t)}{t} = 1 = l n y

l n y = 1 s o y = e^{1} = e

Commented by Tawa1 last updated on 24/Jan/19

God bless you sir

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