Find the horizontal asymptote for `f(x)=(3x^2-1)/(2x-1)` :
A rational function will have a horizontal asymptote of y=0 if the degree of the numerator is less than the degree of the denominator. It will have a horizontal asymptote of `y=a_n/b_n` if the degree of the numerator is the same as the degree of the denominator (where `a_n,b_n` are the leading coefficients of the numerator and denominator respectively when both are in standard form.)
If a rational function has a numerator of greater degree than the denominator, there will be no horizontal asymptote. However, if the degrees are 1 apart, there will be an oblique (slant) asymptote.
For the given function, there is no horizontal asymptote.
We can find the slant asymptote by using long division:
The slant asymptote is `y=3/2x+3/4`
The graph of the function and the asymptote in red: