Question

Discuss the behavioure of the sequence
limit n tends to infinity (1+ 1/n)^n​

Answer 1

Step-by-step explanation:

let y = lim (1+(1/n))ⁿ

apply ln on both sides

ln y = lim ln (1+(1/n))ⁿ , n -> infinity

= n lim ln(1+1/n)) , n->infinity

let x =1/n, as n ->infinity, x->0

ln y = 1/x lim ln (1+x) = lim ln (1+x)/x

x->0 x->0

ln y = lim ln (1+x)

______ as x->0

lim x

using l hospital rule lim f(x)/g(x) =lim f'(x)/g'(x)

d/dx(ln(1+x)) = 1/(1+x)

lny = lim 1/(1+x) / 1 = 1

x->0

ln y =1

y = e¹ =e