As n approaches infinity, find the limit of (n^p)/(e^n) when p0.

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.

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.

anonymous · Answer

$$\lim_{n\to\infty }\frac{n^p}{e^n}$$

anonymous · Answer

you have to visualize l'hopital repeated $p$ times

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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!