The vertical asymptote is found when #f(x)# tends to infinity. #f(x)# normally tends to infinity when the denominator tends to #0#.
So here:
#3x-1=0#
#3x=1#
#x=1/3# is the vertical asymptote.
For the horizontal asymptote, we use the degrees of the numerator and the denominator. Say #m# is the former and #n# the latter. If:
#m>n#, then there is no horizontal asymptote, only a slant.
#m=n#, the horizontal asymptote is at the quotient of the leading coefficient of the numerator and denominator
#m<##n#, the asymptote is at #y=0#.
Here, #m=1# and #n=1#. So #m=n#.
We must divide the leading coefficients of the numerator #(2)# and the denominator (#3#).
#y=2/3# is the horizontal asymptote.
The x-intercept is found when #f(x)=0#. Here,
#(2x)/(3x-1)=0#
#2x=0#
#x=0# is the x-intercept.
The y-intercept is the answer to #f(0)#. Inputting:
#(2*0)/(3*0+1)#
#0/1#
#y=0# is the y intercept.
