Daily Maths Challenges

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submitted 9 months ago by siriusmart to c/dailymaths

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[–] zkfcfbzr 4 points 9 months ago

solution

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

[–] siriusmart 3 points 9 months ago

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

Hint:

spoiler

Use a suitable substitution.

Solution:

spoiler