Question

How do you find the vertical, horizontal or slant asymptotes for `(5x^3-2x^2+1)/(4x^3+2x-7)`?

Answer 1

Vertical : `uarr x = 1. 067 darr`, nearly.
Horizontal : `larr y = 1.25 rarr`

Explanation:

By actual division,

the given function

`y =5/4+(-2x^2-5x/2+35/4)/(4x^3+2x-7)`, giving

an asymptote y = quotient =5/4.

After due analysis of the second graph of the denominator-cubic,

for its zeros,

there is just another asymptote

x = 1.067, nearly.

See the asymptotes-inclusive first graph for the given function.

graph{(y-5/4)(x-1.067)(y-(5x^3-2x^2+1)/(4x^3+2x-7))=0x^2 [-5, 5, -2.5, 2.5]}

graph{4x^3+2x-7 [-20, 20, -10, 10]}