Question

Limits goes to negative INFINITY = (X-9)/ SQUARE ROOT OF (4X^2+3X+2)

Answer 1

horizontal asymptote is found by checking if expression converges to some fixed value as x goes to +infinity or -infinity.

in your case you have such limit y=1/2 (found by differentiation and using l'Hospital's rule)


vertical asymptotes are found at points where expression is not defined. for example we cannot divide by zero or have root of negative number etc.
for those reasons, in your case 4x^2+3x+2 must be greater than zero.
finding roots of the quadratic expression will give you x points where you have vertical asymptotes