Can't seem to figure out this limit

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Homework Statement


I'm trying to do this limit based on a previous thread ( https://www.physicsforums.com/threads/proving-n-x-n-e-x-integrated-from-0-to-infinity.641947/#_=_ )

I got up to the last part of thread where I need to find the limit of:
limit as x approaches infinity of: (-x^(k+1))/e^x

Homework Equations




The Attempt at a Solution


I know that this limit somehow must equal to zero in order to get the right answer, but I did l'Hopital's rule 4 times and it just keeps on going to infinity.

I attached the working out of the whole problem

Really appreciate it if someone could help
 

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Answers and Replies

  • #2
LCKurtz
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If you keep using L'Hospitals rule with a polynomial in the numerator and an exponential in the denominator, the numerator's degree will eventually become 0 while the exponential remains in the denominator.
 
  • #3
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If you keep using L'Hospitals rule with a polynomial in the numerator and an exponential in the denominator, the numerator's degree will eventually become 0 while the exponential remains in the denominator.

I cant seem to get it to work here because the exponent of the polynomail has a degree k. If the degree of the polynomail is a variable constant im not sure how i can get it to zero
 
  • #4
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I cant seem to get it to work here because the exponent of the polynomail has a degree k. If the degree of the polynomail is a variable constant im not sure how i can get it to zero

What's the derivative of any constant?
 
  • #5
pasmith
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Homework Statement


I'm trying to do this limit based on a previous thread ( https://www.physicsforums.com/threads/proving-n-x-n-e-x-integrated-from-0-to-infinity.641947/#_=_ )

I got up to the last part of thread where I need to find the limit of:
limit as x approaches infinity of: (-x^(k+1))/e^x

Homework Equations




The Attempt at a Solution


I know that this limit somehow must equal to zero in order to get the right answer, but I did l'Hopital's rule 4 times and it just keeps on going to infinity.

Keep going. You must apply l'Hopital [itex]k + 1[/itex] times in total before you get a constant in the numerator.

Alternatively, as every term in the series [itex]e^x = \sum_{n=0}^\infty \frac{x^n}{n!}[/itex] is strictly positive when [itex]x > 0[/itex], we have [itex]e^x > \frac{x^{k+2}}{(k+2)!}[/itex] and hence [tex]0 < \frac{x^{k+1}}{e^x} < \frac{(k+2)!x^{k+1}}{x^{k+2}} = \frac{(k+2)!}{x}.[/tex] Now use the squeeze theorem.
 
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  • #6
StoneTemplePython
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Keep going. You must apply l'Hopital [itex]k + 1[/itex] times in total before you get a constant in the numerator.

Alternatively, as every term in the series [itex]e^x = \sum_{n=0}^\infty \frac{x^n}{n!}[/itex] is strictly positive when [itex]x > 0[/itex], we have [itex]e^x > \frac{x^{k+2}}{(k+2)!}[/itex] and hence [tex]0 < \frac{x^{k+1}}{e^x} < \frac{(k+2)!x^{k+1}}{x^{k+2}} = \frac{(k+2)!}{x}.[/tex] Now use the squeeze theorem.


This is how I'd do it, as I tend to think l'Hopital is a rather unintuitive power-tool that should be used as a last resort.

At a minimum, I'd change it to

##0 \leq \frac{x^{k+1}}{e^x} \leq ...##

though, otherwise that strict inequality would seem to cause problems, as in the limit we have ## 0 \lt 0##
 

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