Very lost with l'Hospital's rule

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Homework Help Overview

The problem involves evaluating the limit as x approaches infinity of the expression \((\frac{x}{x+1})^x\) and identifying the type of indeterminate form present.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of l'Hospital's rule, particularly in the context of different indeterminate forms such as 0/0 and ∞/∞. There is a suggestion to transform the expression into a logarithmic form to facilitate evaluation. Some participants question whether the expression is in the form of 1^∞ and explore how to rearrange it into a suitable form for applying l'Hospital's rule.

Discussion Status

The discussion is active, with participants providing hints and guidance on how to approach the limit. There is a recognition of the need to take the logarithm first and then apply l'Hospital's rule, although there is no explicit consensus on the final steps or outcomes.

Contextual Notes

Participants note the challenge of dealing with the indeterminate form and the requirement to rearrange the expression appropriately for analysis. The context includes a quiz setting, which may impose additional constraints on the discussion.

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Homework Statement


Evaluate lim x[tex]\rightarrow[/tex] infinity of ([tex]\frac{x}{x+1}[/tex])[tex]^{}x[/tex], 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.
 
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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.
 
  • #10
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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