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

Apr 29th 2011, 10:43 AM

HallsofIvy

Quote:

Originally Posted by prasum

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

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.

Apr 29th 2011, 11:25 AM

prasum

Quote:

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.

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