What is the limit of # f(x) = (2x+5)/abs(3x-4)# as x goes to infinity? Calculus Limits Limits at Infinity and Horizontal Asymptotes 1 Answer Sasha P. Oct 18, 2015 #2/3 vv -2/3#, see the explanation. Explanation: #|3x-4|=3x-4# for #3x-4>=0, 3x>=4, x>=4/3# #|3x-4|=-3x+4# for #3x-4<0, 3x<4, x<4/3# #lim_(x->+oo) (2x+5)/|3x-4| = lim_(x->+oo) (2x+5)/(3x-4) = lim_(x->+oo) (2+5/x)/(3-4/x) = 2/3# #lim_(x->-oo) (2x+5)/|3x-4| = lim_(x->-oo) (2x+5)/(-3x+4) = lim_(x->-oo) (2+5/x)/(-3+4/x) = -2/3# Note: #5/x->0# when #x->oo vv x->-oo# #4/x->0# when #x->oo vv x->-oo# Answer link Related questions What kind of functions have horizontal asymptotes? How do you find horizontal asymptotes for #f(x) = arctan(x)# ? How do you find the horizontal asymptote of a curve? How do you find the horizontal asymptote of the graph of #y=(-2x^6+5x+8)/(8x^6+6x+5)# ? How do you find the horizontal asymptote of the graph of #y=(-4x^6+6x+3)/(8x^6+9x+3)# ? How do you find the horizontal asymptote of the graph of y=3x^6-7x+10/8x^5+9x+10? How do you find the horizontal asymptote of the graph of #y=6x^2# ? How can i find horizontal asymptote? How do you find horizontal asymptotes using limits? What are all horizontal asymptotes of the graph #y=(5+2^x)/(1-2^x)# ? See all questions in Limits at Infinity and Horizontal Asymptotes Impact of this question 2303 views around the world You can reuse this answer Creative Commons License