How to find limit of a sequence of functions $f_n(x)=\frac{x^n e^x} {n+1}$? $$\lim_{n\to \infty} \frac{x^n e^x}{n+1}$$
I have no idea how to evaluate this limit. I thought maybe I should rewrite $e^x$ using $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ but I am not sure whether I can do that or whether it would even help. If it is possible to evaluate the limit without L'Hospital's rule that would be the prefered way but I actually cannot see how L'Hospital would help with this problem.
We get the same result if we compute
$$e^x\lim_{n\to+\infty}\frac{x^n}{n}$$
If $|x|\le 1$, it gives zero.
If $ x>1$, we write it as
$$\lim_{n\to+\infty}e^{n(\ln(x)-\frac{\ln(n)}{n})}=+\infty$$
If $x<-1,$ the limite doesn't exist, since for even indices, we find $+\infty$ ,and for odd ones, it gives $-\infty$.