What is the limit of f(x) = (2x+5)/abs(3x-4) as x goes to infinity?

1 Answer
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