Use L'Hopital's rule to evaluate the limit

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Homework Help Overview

The problem involves evaluating the limit of the expression (x/(x+1))^x as x approaches infinity, with a focus on applying L'Hopital's rule. The context is rooted in calculus, particularly in the study of limits and derivatives.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss transforming the limit into a logarithmic form to facilitate the application of L'Hopital's rule. There are attempts to derive expressions and evaluate derivatives, but some participants express confusion about the effectiveness of their approaches. Questions arise regarding the necessity of using L'Hopital's rule and the proper application of derivatives.

Discussion Status

The discussion is ongoing, with various participants sharing their attempts and reasoning. Some guidance has been offered regarding the use of logarithmic properties and the application of L'Hopital's rule, but there is no explicit consensus on the correct approach or outcome at this stage.

Contextual Notes

There is mention of indeterminate forms arising during the evaluation process, and some participants highlight the need for clarity on the application of derivatives in this context. The original poster's repeated attempts indicate a struggle with the problem setup and the methods being employed.

physstudent1
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Homework Statement



Use L'Hopital's rule to evaluate the limit:

lim (x/(x+1))^x
x>infinite

Homework Equations


The Attempt at a Solution



I put it into a logarithm first to make it the limit of xln(x/(x+1)) then I took the derivative and got [(1/x)-(1/(x+1))]/[x^-2]
but its still indeterminate and after taking the 2nd derivative it was as well
 
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my guess would be you have to use some log properties like log(a/b) = log(a) - log(b)
 
i did do that to get to the derivative but the derivatives didn't help
 
You forgot to use the power rule since there are 2 functions

This is what I got (could be wrong)

= x(logx - log (x+1)
= logx - log(x+1) + x((1/x)-(1/x-1)) <---simplify
= logx - log(x+1) - (1/x-1)

I believe you need to take the second derivative and you should get the limit to be 0 (it's been about a year since I did diffrential calc)
 
Why use L'Hopital? Just look at its reciprocal function. Trivial.
 
Kummer said:
Why use L'Hopital?

To get more experience using it?
 
physstudent1 said:

Homework Statement



Use L'Hopital's rule to evaluate the limit:

lim (x/(x+1))^x
x>infinite

Homework Equations





The Attempt at a Solution



I put it into a logarithm first to make it the limit of xln(x/(x+1)) then I took the derivative and got [(1/x)-(1/(x+1))]/[x^-2]
but its still indeterminate and after taking the 2nd derivative it was as well

Why the derivative? L'Hospital's rule usually involves taking two derivatives separately! Write the logarithm as
[tex]\frac{ln(x)- ln(x+1)}{x^{-1}}[/tex]
and apply L'Hopital's rule to that.
 

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