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A very difficult limit

  • #1

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.
 

Answers and Replies

  • #2
Orodruin
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Use that ##a^x = \exp(x \ln(a))## and start expanding the log in powers of ##1/x##.
 
  • #3
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.
 
  • #4
Orodruin
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I suggest you start the way I suggested. The rest will follow.
 
  • #5
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.
 
  • #6
Orodruin
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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.
 
  • #7
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.
 
  • #8
vela
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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.
 
  • #9
Orodruin
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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?
 
  • #10
Ok so here is my try
246083


Now shall i use L'Hospitale from here or something else.
 
  • #11
Orodruin
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I suggest that you use that ##\ln(a/b) = - \ln(b/a)## and do some rewriting of that logarithm.
 
  • #12
I suggest that you use that ##\ln(a/b) = - \ln(b/a)## and do some rewriting of that logarithm.
Thanks i got the answer.
 
  • #13
Orodruin
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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.
 
  • #14
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
246095

Here is my solution.Is it ok?
 

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