Let #f(x)=(P(x))/(Q(x))# be a rational function, meaning that #P(x)# and #Q(x)# are polynomials. If the degrees (highest powers) of #P(x)# and #Q(x)# are the same, then #lim_{x-> pm infty}f(x)# exists and equals the ratio of the coefficients of the highest powers of #P(x)# and #Q(x)#.
For the example at hand, the numerator expands to
#(3x+4)(x-1)(2x+1)=(3x^2+x-4)(2x+1)#
#=6x^3+2x^2-8x+3x^2+x-4=6x^3+5x^2-7x-4#.
Therefore,
#lim_{x-> pm infty} ((3x+4)(x-1)(2x+1))/(ax^3+x-4)=6/a# when #a!=0#.'
When #a=0#, then the numerator has a higher degree than the denominator, so the limit does not exist.
When the denominator has a higher degree than the numerator, then the limit is zero.
