Very lost with l'Hospital's rule

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Homework Statement


Evaluate lim x\rightarrow infinity of (\frac{x}{x+1})^{}x, state explicitly the type of the indeterminate form.



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The Attempt at a Solution


I somewhat understand how to use l'Hospital's rule when the form is 0/0, but the inf/inf throws me off completely.
f'(x) = (x/(x+1))^x *ln(x/(x+1))
but from there or maybe the start I'm lost.
 
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whether its 0/0 or inf/inf you proceed the same way...take the derivative of the top function and the derivative of the bottom function and evaluate the limit again
 
First you need to change the expression to a 0/0 form. Hint: Take the log first, then try to rearrange that into a 0/0 form.
 
I should have posted this sooner, but isn't it in the form 1^inf ?
 
Yes, that's why you need to take the log first. Find the limit of the logarithm, then exponentiate to get the limit of the original expression.
 
so you would get e^limx->inf of xln(x/(x+1)) and then what?
 
Uh, rearrange xln(x/(x+1)) into a 0/0 form and apply l'Hopital to find the limit.
 
ok I got that much, and I got x/(1/ln(x/(x+1))), but I'm lost as to evaluate that lim with l'Hospitals, would it be (1) / (1/ln(x/(x+1)))', if so I haven't the slightest as to how to get that derivative.
 
It's a lot easier to write it as ln(x/(x+1))/(1/x). Use the chain rule to do the ln part.
 
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hey thanks for all your help guys, i got the answer (e^-1) finally and it was for a quiz so i appreciate it.
 
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