[2024/05/08] Definition of e^x

zkfcfbzr, 1 month ago

solutionlim (n → ∞) (1 + x/n)^n = e^( lim (n → ∞) ln( (1 + x/n)^n ) ) = e^( lim (n → ∞) n * ln(1 + x/n) ) = e^( lim (n → ∞) ln(1 + x/n) / (1/n) ) = e^( lim (n → ∞) (1/(1 + x/n) * -x/n^2) / (-1/n^2) ) → L’Hôpital = e^( lim (n → ∞) x / (1 + x/n) ) = e^( x / (1 + 0) ) = e^x I’m at least 60% sure this proof isn’t somehow circular

siriusmart, 1 month ago

There seemed to be more than one ways to prove this.

Hint:

spoilerUse a suitable substitution.

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Solution:

spoilergmtex.siri.sh/…/2024-05-08_e^x-definition.htmlhttps://lemmy.world/pictrs/image/27caff35-d892-4083-b262-b8b6d59e9937.png