Can l'Hôpital's Rule Help Determine the Limit of an Improper Integral?

In summary, the conversation discusses proving an indeterminate form involving a limit as x approaches 0 and the use of l'Hopital's rule and induction. The conversation also mentions the substitution x=e^-v and comparing exponential and polynomial functions. Ultimately, l'Hopital's rule and induction are the recommended methods for solving this problem.
  • #1
trap
53
0
Proving an indeterminate form

Prove for all positive integers n that [tex] \lim_{x\rightarrow 0}x({lnx})^n=0 [/tex]

Thanks for any help.
 
Last edited:
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  • #2
That's not an integral.

Do you know l'Hopital's rule? Combine it with induction.
 
  • #3
oh...i didn't think of induction
i kept doing it by l'hopital's rule and it came out infinity/infinity all the time.
Thanks for the advice
 
  • #4
It's futile to use L'Hôpital's rule (you can't get a reasonable expression for

[tex] \frac{d^{k}(\ln x)^{n}}{dx^{k}} [/tex]

)

Do a substitution:

[tex] x=e^{-v} [/tex]

The result is immediate.It's like comparing exp & a finite polynomial.Since "n" is fixed,the factor [itex] (-1)^{n} [/itex] bears no relevance...

Daniel.
 
  • #5
L'Hôpital's rule + induction works fine for me... just like Jameson said.
 
  • #6
dextercioby said:
It's like comparing exp & a finite polynomial.

and how do you know what happens in this comparison if you aren't familiar with it? enter l'Hôpital... :tongue2:
 

What is an improper integral?

An improper integral is an integral where one or both of the limits of integration are infinite or the integrand has an infinite discontinuity within the interval of integration.

Why is it important to prove an improper integral?

Proving an improper integral is important because it ensures that the integral is well-defined and converges to a finite value. This is crucial in many applications, such as in physics and engineering, where improper integrals are commonly used.

How do you prove that an improper integral converges?

In order to prove that an improper integral converges, you must evaluate the limit as the upper and lower bounds of integration approach infinity. If the limit exists and is finite, the improper integral is said to converge.

What happens if an improper integral does not converge?

If an improper integral does not converge, it is said to diverge. This means that the integral does not have a finite value and cannot be evaluated. In this case, it is not possible to use the integral in calculations or applications.

What are some common techniques used to prove an improper integral?

Some common techniques used to prove an improper integral include the comparison test, the limit comparison test, and the use of the Cauchy principal value. These techniques involve comparing the given integral to known integrals and using limits to evaluate the integral.

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