Question

How do you find vertical, horizontal and oblique asymptotes for `f(x)= (2x^3+11x^2+5x-1)/(x^2+6x+5)`?

Answer 1

Vertical : `x = -1 and x = - 5`.
Slant: `y = 2x - 1`.
See Socratic graph for the curve and the asymptotes.

Explanation:

`y = f(x)=2x-1+(x-4)/((x+1)(x+5)`

y = quotient = `2x-1` gives the slant asymptote, with slope 2.

The factors ( x + 1 ) and ( x + 5 ) in the denominator of the remainder

equated to 0 give the vertical asymptotes.

The ad hoc graphs are not in uniform as-is scales.

They serve the purpose of revealing asymptoticity of the three

asymptotes, with respect to the three branches of the graphs, taken

in pairs..

graph{(y(x+1)(x+5)-2x^3-11x^2-5x+1)(x+5)(y-2x+1)(x+1)=0 [-10, 10, -30, `30]}

graph{(y(x+1)(x+5)-2x^3-11x^2-5x+1)(x+5)(y-2x+1)(x+1)=0 [-40, 40, -82.5, 57.5]}