Question

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

Answer 1

`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`