Question

How do you find vertical, horizontal and oblique asymptotes for `(x+4)/(3x^2+5x-2)`?

Answer 1

vertical asymptotes x = -2 , x `= 1/3`
horizontal asymptote y = 0

Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation/s set the denominator equal to zero.

solve : `3x^2 + 5x - 2 = 0 → (3x-1)(x+2) = 0`

`rArr x = - 2 , x = 1/3" are the asymptotes "`

Horizontal asymptotes occur as `lim_(xto+-oo) f(x) to 0`

When the degree of the numerator < degree of the denominator , the equation is always y = 0 .

Oblique asymptotes occur when the degree of the numerator > degree of the denominator . This is not the case here hence there are no oblique asymptotes.
graph{(x+4)/(3x^2+5x-2) [-10, 10, -5, 5]}