I couldn't find the following limit:

lim*x*→∞*x*((1+1*x*)*x*−*e*).

Definitely, it has to be through the L'Hospital's rule. We know that

lim*x*→∞(1+(1/*x*))*x*=*e*. So, I wrote the above expression as

lim*x*→∞(1+1*x*)*x*−*e*1/*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