How do you find the vertical, horizontal or slant asymptotes for #f(x) = (x^3-8)/(x^2-5x+6)#?
Horizontal asymptotes: None
In order to work out whether a rational function,
In this case:
Using any method to solve this equation tells us that
Horizontal asymptotes occur when the polynomial of the denominator of a rational function has a higher degree than the polynomial of the numerator. If so, then the
In this function, this doesn't occur, so there are no horizontal asymptotes.
Slant asymptotes occur when the polynomial of the denominator of a rational function has a lower degree than the polynomial of the numerator. In order to find our slant asymptote, we must divide the numerator by the denominator.
If we divide the numerator by the denominator, we get the slant asymptote as
And here's your graph plotted on Desmos. (Although it seems like the graph crosses the first horizontal asymptote, the graph is actually undefined for that part.)