limits

lim ((1+x)^(1/x)-e)/x can yu do it by l,hospital rule
x\to0

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Originally Posted by prasum

lim ((1+x)^(1/x)-e)/x can yu do it by l,hospital rule
x\to0

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You can but it get really nasty! Are you required to us L'Hopital's rule?
It's much easier if you recognize that letting 1/x= p gives
lim(p-> infinity) (1+ 1/p)^p = e. This turns out to be, essentially, a derivative.

Quote:

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Originally Posted by HallsofIvy

You can but it get really nasty! Are you required to us L'Hopital's rule?
It's much easier if you recognize that letting 1/x= p gives
lim(p-> infinity) (1+ 1/p)^p = e. This turns out to be, essentially, a derivative.

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i have done by substituting (1+x)^1/x=e^((1/x)(ln(1+x))) so appling expansion of e we get limit as -e/2 can yu please suggest graphical approach