lim (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

Use a suitable substitution.

https://gmtex.siri.sh/fs/1/School/Extra/Maths/Qotd solutions/2024-05-08_e^x-definition.html