global behavior of functions

back in pre-calculus i was taught that to find the slant asymptote of a rational function, i have to divide the top by the bottom and whatever i get is my asymptote. Now in calculus, i am taught to find the global behavior that i divide the top and bottom by the highest power in the bottom and let x become extremely large. if the higest power in the numerator is higher than the highest power in the denominator, the global behavior would be a function of some sort. as x approaches +/- infinite, the function approaches some other function which is the asymptote.

So what's the difference between the global behavior and whatever asymptote the rational function approaches? for some rational functions such as (5x^3 - 2x^2 + 1) / (1 - 3x), if i divide top by bottom (asymptote) and if i divide top and bottom by highest power in denominator (global behavior) i get two different answers. but aren't they the same thing?