TITLE: What is the value of $\displaystyle\lim_{R\to\infty} {\int_R^{\infty}{r^ne^{-r^2/2}dr}\over{R^{n-1}e^{-R^2/2}}}$
QUESTION [5 upvotes]: What is the value of $\displaystyle\lim_{R\to\infty} {\int_R^{\infty}{r^ne^{-r^2/2}dr}\over{R^{n-1}e^{-R^2/2}}}$, where n is a positive integer? I don't know how to integrate the numerator.

REPLY [4 votes]: The quotient we are investigating is of the form $0/0$, and so we may apply L'Hopital's rule. We differentiate the numerator using the second fundamental theorem of calculus and the denominator using the product rule to get that our limit is the same as
$$
\lim_{R \to \infty} \frac{-R^ne^{-R^2/2}}{(n-1)R^{n-2}e^{-R^2/2} - R^n e^{-R^2/2}}.
$$
Now you can factor $R^ne^{-R^2/2}$ out of numerator and denominator to get
$$
\lim_{R \to \infty} \frac{-1}{(n-1)R^{-2} - 1}
$$
which is clearly 1.