Use L'Hopital's Rule to evaluate the limit

In summary, the conversation is about using L'Hopital's Rule to evaluate the limit of a function, specifically lim x-infinity of (lnx) ^(2/x). The answer is 1, but the function initially seemed to be in a form that could not be solved using L'Hopital's Rule. The group discusses how to convert the function into an indeterminate form of 0/0 or inf/inf in order to apply L'Hopital's Rule. The conversation ends with a suggestion to convert the function into f(x)/(1/g(x)) to get an appropriate indeterminate form.
  • #1
Shay10825
338
0
Hi. Use L'Hopital's Rule to evaluate the limit.

lim x-infinity of (lnx) ^(2/x)

The answer is 1.
I kept taking the derivative but it seemed like I was going around in circles. Any help would be appreciated.

Thanks
 
Physics news on Phys.org
  • #2
L'Hopital's Rule does not apply to the function as written. What were your first steps?
 
  • #3
I found the derivative of the function and it was really ugly. I don't know if it is correct.

[ [-2(lnx)^(2/x -1)][lnx*ln(lnx) -1] ]/x^2
 
  • #4
As Diane said, you can't apply L'Hopital's rule to this function, at least not in this form.
It is possible however to alter the function so it becomes something where you can use L'Hopital. You need to convert it to an undeterminate form of 0/0 or inf/inf.
 
  • #5
How can I convert it?
 
  • #6
Well, what you have here is a case of f(x)^g(x) which yields the indeterminate form [itex]\infty ^0[/itex].

You can convert it to another indeterminate form by doing [itex]\exp \left( {\ln \left( {f\left( x \right)^{g\left( x \right)} } \right)} \right) = \exp \left( {g\left( x \right)\ln \left( {f\left( x \right)} \right)} \right)[/itex].

Then you have something of the form f(x)g(x) which gives a new indeterminate form [itex]\infty \cdot 0[/itex].

Finally, you can convert this to f(x)/(1/g(x)) (or the other way arround) to get either 0/0 or inf/inf so that you can use L'Hopital.
 

What is L'Hopital's Rule?

L'Hopital's Rule is a mathematical tool used to evaluate limits of indeterminate forms, where the numerator and denominator both approach zero or infinity. It states that if the limit of a function f(x) divided by g(x) is in an indeterminate form (0/0 or ∞/∞), then the limit of the derivative of f(x) divided by the derivative of g(x) will equal the original limit.

When should I use L'Hopital's Rule to evaluate a limit?

L'Hopital's Rule should only be used when the limit is in an indeterminate form. This means that the limit of the function f(x) divided by g(x) is either 0/0 or ∞/∞. If the limit is not in an indeterminate form, then L'Hopital's Rule is not applicable and other methods should be used to evaluate the limit.

How do I use L'Hopital's Rule to evaluate a limit?

To use L'Hopital's Rule, you must first determine if the limit is in an indeterminate form. If it is, then take the derivative of the numerator and denominator separately. Then, evaluate the limit of the derivative of the numerator divided by the derivative of the denominator. This will give you the same value as the original limit.

What are some common mistakes when using L'Hopital's Rule?

One common mistake is using L'Hopital's Rule when the limit is not in an indeterminate form. Another mistake is not taking the derivatives correctly, which can lead to incorrect results. It is also important to ensure that the limit of the derivative of the numerator divided by the derivative of the denominator is actually equal to the original limit, as this may not always be the case.

Are there any limitations to using L'Hopital's Rule?

Yes, there are limitations to using L'Hopital's Rule. It can only be used to evaluate limits of indeterminate forms, so it is not applicable to all limits. Additionally, it may not always give the correct result, so it is important to check the answer using other methods as well.

Similar threads

Replies
1
Views
608
  • Calculus and Beyond Homework Help
Replies
17
Views
486
  • Calculus and Beyond Homework Help
Replies
23
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
653
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
Back
Top