Question

As n approaches infinity, find the limit of (n^p)/(e^n) when p>0.

I keep getting an indeterminate form. I don't think I can use L'Hopital's rule, or can I? I'm not sure how to tackle this problem.

Answer 1

\[\lim_{n\to\infty }\frac{n^p}{e^n}\]

Answer 2

you have to visualize l'hopital repeated \(p\) times

Answer 3

the denominator stays \(e^n\) but eventually the numerator is a constant, namely \(p!\)

Answer 4

making the limit zero, telling you that in the long run, an exponential grows faster than any polynomial

Answer 5

thank you so much. i dont know why my brain does not understand this stuff by itself! when people word things with deatil, everything somehow makes sense. anyways, thanks again!