Suppose you wish to find the asymptotes of the following:
2x/(3x + 2)
To find the vertical asymptote, find where the denominator is undefined for x. Set the denominator equal to zero.
3x + 2 = 0
x = -2/3
Therefore the vertical asymptote is the line x = -2/3
To find the horizontal asymptote, divide the coefficients of the x term (numerator/denominator). Notice the coefficients are 2 and 3.
The horizontal aymptote is the line y = 2/3.
The graph will approach but never touch the asymptotes.