What is the horizontal asymptote of the rational function f (x) = 3x / (2x - 1) ?

Step-by-step explain

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:

(3x^2-1) / (2x-1) = (2x-1) (3/2x+3/4 - (1/4) / (2x-1))

The slant asymptote is y=3/2x+3/4

y = 3/2.

Step-by-step explanation:

This is the ratio of the coefficients of the terms in x of highest degree of the function:

y = 3/2.