# A very difficult limit

## Homework Statement:

Problem-:
Lim x(1/e-(x/(1+x))^x)
Where x tends to infinity

## Homework Equations:

L'Hospitale
I simplified somewhat and got (1/e-(1-x/(1+x))x)/(1/x)
So i cant find that it is 0/0 form so tried by applyying L'Hospitale,But it just became complicated.So need help.

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Orodruin
Staff Emeritus
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Use that $a^x = \exp(x \ln(a))$ and start expanding the log in powers of $1/x$.

Use that $a^x = \exp(x \ln(a))$ and start expanding the log in powers of $1/x$.
Can you explain a little more.I think my a here is $x/(1+x)$.But what will i do with the x that is outside the bracket.

Orodruin
Staff Emeritus
Homework Helper
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I suggest you start the way I suggested. The rest will follow.

I suggest you start the way I suggested. The rest will follow.
But i didn't got what you said.Please explain a little more.

Orodruin
Staff Emeritus
Homework Helper
Gold Member
You were correct in identifying $a$, now what do you get when you use that on your expression? I need to see how far you have gotten to be able to help you.

You were correct in identifying $a$, now what do you get when you use that on your expression? I need to see how far you have gotten to be able to help you.
Ok after then i opened the expansion of ln($x$/$(1+x)$).But i got stuck afterwards.

vela
Staff Emeritus
Homework Helper
Ok after then i opened the expansion of ln($x$/$(1+x)$).But i got stuck afterwards.
It's really unhelpful to say, "I did X and then I got stuck." What you mean by X and what we think you meant by X could be entirely different. Please show your work.

Orodruin
Staff Emeritus
Homework Helper
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Please show the entire expression that you have and what your thoughts are about it, i.e., why are you stuck?

Ok so here is my try

Now shall i use L'Hospitale from here or something else.

Orodruin
Staff Emeritus
Homework Helper
Gold Member
I suggest that you use that $\ln(a/b) = - \ln(b/a)$ and do some rewriting of that logarithm.

I suggest that you use that $\ln(a/b) = - \ln(b/a)$ and do some rewriting of that logarithm.

Orodruin
Staff Emeritus
Homework Helper
Gold Member