Why does (2^n)/(n^4) approach infinity as n approaches infinity?

[UrbanXrisis]

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.

 

[Astronuc]

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

 

[Tide]

It might help to compare logarithms of the numerator and denominator.

 

[HallsofIvy]

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.)