Simplified Limit Calculation for (1-e^(1-x/(1+x))x)/(1/x)

[Physics lover]

Homework Statement

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

Relevant Equations

L'Hospitale

I simplified somewhat and got (1/e-(1-x/(1+x))^x)/(1/x)
So i can't find that it is 0/0 form so tried by applyying L'Hospitale,But it just became complicated.So need help.

 

[Orodruin]

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

 

[Physics lover]

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]

I suggest you start the way I suggested. The rest will follow.

 

[Physics lover]

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]

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.

 

[Physics lover]

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]

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]

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

 

[Physics lover]

Ok so here is my try

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

 

[Orodruin]

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

 

[Physics lover]

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

Thanks i got the answer.

 

[Orodruin]

Thanks i got the answer.

Would you mind sharing? Forum rules prevents others from posting solutions until the questioner has shown their solution and it may be of use to others finding this thread.

 

[Physics lover]

Would you mind sharing? Forum rules prevents others from posting solutions until the questioner has shown their solution and it may be of use to others finding this thread.

Ok

Here is my solution.Is it ok?

 

[Lifeforbetter]

Ok
View attachment 246095
Here is my solution.Is it ok?

what theory? What chapter of math?