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# Limit derivation

## Question

lim((1-1/x)^x) [x→ ∞] = (1-1/e)

How would one derive this limit?

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lim((1-1/x)^x) [x→ ∞] = 1 not (1-1/e)

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

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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.

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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

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