Help!!! I couldn't evaluate this limit

[GiaBao]

[math]\lim_{x\to\infty}{x \left[e-\left(1+\frac{1}{x}\right)^x\right]}[/math]I tried to use [imath]e^{\ln f(x)}[/imath] and L'Hospital rule but both didn't work.
I think this limit is similar to [math]\lim_{x\to0}{\frac{e-(1+x)^\frac{1}{x}}{x}}[/math][math][/math] but not, it's hard much more this one.
I'm very grateful if you could help me

 

[Dr.Peterson]

[math]\lim_{x\to\infty}{x \left[e-\left(1+\frac{1}{x}\right)^x\right]}[/math]I tried to use [imath]e^{\ln f(x)}[/imath] and L'Hospital rule but both didn't work.
I think this limit is similar to [math]\lim_{x\to0}{\frac{e-(1+x)^\frac{1}{x}}{x}}[/math][math][/math] but not, it's hard much more this one.
I'm very grateful if you could help me

These two limits are practically the same! You can get the latter from the former by just making a change of variables, letting [imath]y=\frac{1}{x}[/imath].

So what limit do you get for the latter?

 

[blamocur]

So what limit do you get for the latter?

The latter yields to L'Hospital rule or two

 

[blamocur]

I tried to use [imath]e^{\ln f(x)}[/imath] and L'Hospital rule but both didn't work.
I think this limit is similar to [math]\lim_{x\to0}{\frac{e-(1+x)^\frac{1}{x}}{x}}[/math][math][/math] but not, it's hard much more this one.
I'm very grateful if you could help me

Can you figure out the derivative of [imath](1+x)^\frac{1}{x}[/imath] ?