Prologue:On understanding the use of binomial theorem to find asymptotes of a real valued function
So i understood how to obtain non linear asymototes of a function if any. So how do you know if the function has more than one non linear asymptote? I mean look at the part of the graph "in the 3rd quadrant". How do i know it wont have any non linear asymptote "there"?
Here is the reference Question used on all my posts:
y =$$(x^3+2x+9)$$/$$(sqrt(4x^2+3x+2))$$
My asymptote obtained using long division
y=(x^2/2) -(3x/16) +(9/128)
Graph: 
Red line: Original Function
Black line: Asymptote