4 replies to this topic

[vinay.singh84]

lim((1-1/x)^x) [x→ ∞] = (1-1/e)

How would one derive this limit?

[Rob_Gandy]

Spoiler for I would think

lim((1-1/x)^x) [x→ ∞] = 1 not (1-1/e)

[k-man]

Spoiler for it's neither

The limit is neither 1 nor 1-1/e. It's 1/e.

[vinay.singh84]

Sorry messed up the OP.
lim(1-(1-1/x)^x) [x→ ∞] = (1-1/e) or lim((1-1/x)^x) [x→ ∞] = 1/e

The main question of how it is derived stlil stands.

[nitinjain92]

lim x->∞ ( 1 - 1 / X )^X

(can be written also as )

= lim x->∞ e^{( X . log( 1 - 1/X ) )}
( expansion of log ( 1 - 1/X) ) is -1*( 1/x + 1/2x² + 1/3x³ ..... so on)

put this in equation we get



= lim x->∞ e^{( -X *(1/x + 1/2x^2 + 1/3x^3 ..... so on ) )}
=lim x->∞ e^{( -1 *(1 + 1/2x + 1/3x^2 ..... so on ) )}
applying limit we get

answer = e ^-1