Need some clarification on this limit

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Homework Help Overview

The discussion revolves around the limit of the function (2^n)/(n^4) as n approaches infinity. Participants are examining why the limit is stated as infinity in the textbook, despite the intuition that the denominator grows faster than the numerator.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants are questioning the behavior of the numerator and denominator as n approaches infinity, with some suggesting the use of l'Hôpital's rule and comparisons of growth rates. Others propose examining logarithmic comparisons to clarify the limit's behavior.

Discussion Status

The discussion is active, with participants exploring different perspectives on the limit. Some have offered insights into the growth rates of the numerator and denominator, while others are considering alternative methods to analyze the limit without reaching a consensus.

Contextual Notes

There is a repeated emphasis on the perceived growth rates of the numerator and denominator, with participants expressing confusion about the limit's outcome based on their understanding of infinity in this context.

UrbanXrisis
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I need some clarification on this limit:

lim n-->infinity of (2^n)/(n^4)

since the bottom of the function reaches infinity way faster than the top, why isn't the limit zero? my book says that it is infinity.
 
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I believe the numerator goes to infinity faster.

Take n=100, the numberator is 2100 which would be 1625, while the denominator is 1004 = 108.

I believe by applying l'Hôpital's rule, differentiate the numerator and denominator four-fold, the denominator becomes 4!, while the numerator is still factor of 2n
 
It might help to compare logarithms of the numerator and denominator.
 
UrbanXrisis said:
I need some clarification on this limit:
lim n-->infinity of (2^n)/(n^4)
since the bottom of the function reaches infinity way faster than the top, why isn't the limit zero? my book says that it is infinity.

Because the top of the function reaches infinity way faster than the bottom!

(Even 1.00001^n eventually increases faster than n^10000.)
 

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