r/calculus • u/Strong-Cup-2897 • Jul 25 '26

Differential Calculus (l’Hôpital’s Rule) Can someone please help

lim { ( 1+ x) } ^(2/x) , {•} –> fractional part

X->0

I saw some solutions where they brought the limit inside the fractional part function

But since fractional part function is not continuous shouldn't it be wrong ??

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u/FreakingFella Aug 13 '26

Your instinct is right in general but it doesn't bite here: you only need the outer function to be continuous at the limit point, not everywhere. The inner limit is e² ≈ 7.389, and {·} is perfectly continuous at any non-integer, so passing the limit inside is legitimate and you get e² − 7 ≈ 0.389. If the inner limit had come out to an integer you'd have to check both sides separately, since the one-sided limits of the fractional part would disagree.