I couldn't find the following limit:
limx→∞x((1+1x)x−e).Definitely, it has to be through the L'Hospital's rule. We know that limx→∞(1+(1/x))x=e. So, I wrote the above expression as
limx→∞(1+1x)x−e1/x.Both numerator and denominator tend to zero as x tends to infinity. Here comes the L'Hospital's rule. I applied the L'Hospital's rule twice , but I got the limit equal to infinity after the second time which is clearly wrong. In the book, it says that the limit should be −e/2.
Any help please? Also, in case someone managed to solve it, please tell me which numerator and denominator did you apply your L'Hospital's rule to in both cases? Thanks