Applying l'Hospital's rule to find a limit

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

Re: Applying l'Hospital's rule to find a limit

Quote:

Originally Posted by

**skybluesea2010** 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

Sorry but this is unreadable.

Re: Applying l'Hospital's rule to find a limit

Quote:

Originally Posted by

**skybluesea2010** 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

Presumably this means:

$\displaystyle \lim_{x\to \infty}\left\{ x\left[ \left(1+\frac{1}{x}\right)^x-e \right] \right\}$

CB

Re: Applying l'Hospital's rule to find a limit

Exactly, this that's the expression i want to find its limit when x tends to infinity

Re: Applying l'Hospital's rule to find a limit

Quote:

Originally Posted by

**CaptainBlack** Presumably this means:

$\displaystyle \lim_{x\to \infty}\left\{ x\left[ \left(1+\frac{1}{x}\right)^x-e \right] \right\}$

CB

Quote:

Originally Posted by

**skybluesea2010** Exactly, this that's the expression i want to find its limit when x tends to infinity

Then you apply L'Hopital's rule to:

$\displaystyle \lim_{x\to \infty} \frac{ \left(1+\frac{1}{x}\right)^x-e }{(\frac{1}{x})}$