L'Hôpital's rule for limit of x−(x+2)e^(1/x) as x→∞

[Redoctober]

Homework Statement

I know it needs L'Hopital but i can't get it to the indeterminant form

Limit (x-(x+2)e^(1/x)) ((inf - inf ))
x-->inf

I reached until here and then i got stuck

(1-((x+2)e^(1/x))/x)/(1/x) (1 -inf/inf)/0

 

[Curious3141]

Separate limit into $\lim_{x \rightarrow \infty} x(1 - e^{\frac{1}{x}}) - 2e^{\frac{1}{x}}$.

Second term is easy.

First term is indeterminate of the form infinity*0, so reexpress it as $\frac{1 - e^\frac{1}{x}}{\frac{1}{x}}$, then let $y = \frac{1}{x}$ and take the limit as $y \rightarrow 0$ using L' Hopital's Rule.

 

[Redoctober]

Oh i see :D ! Thanks a lot :)