Asymptotic tail properties of student's t-distribution

We investigate the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of Student's t-distribution with nu > 0 degrees of freedom (t(nu) for short) for nu tending to infinity when the argument x = x(nu) of the pdf (cdf) depends on and tends to +/-infinity (-infinity). To this end, we consider the ratio of the pdf's (cdf's) of the t(nu)- and the standard normal distribution. Depending on the choice of the argument x(nu), the pdf-ratio (cdf-ratio) tends to 1, a fixed value greater than 1, or to infinity. As a byproduct, we obtain a result for Mill' ratio when x(nu) -> -infinity.