Very lost with l'Hospital's rule

[coverticus]

Homework Statement

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

Homework Equations

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.

 

[Midy1420]

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

 

[Dick]

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.

 

[coverticus]

I should have posted this sooner, but isn't it in the form 1^inf ?

 

[Avodyne]

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.

 

[coverticus]

so you would get e^limx->inf of xln(x/(x+1)) and then what?

 

[Dick]

Uh, rearrange xln(x/(x+1)) into a 0/0 form and apply l'Hopital to find the limit.

 

[coverticus]

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.

 

[Dick]

It's a lot easier to write it as ln(x/(x+1))/(1/x). Use the chain rule to do the ln part.

 

[coverticus]

hey thanks for all your help guys, i got the answer (e^-1) finally and it was for a quiz so i appreciate it.