Question

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

Answer 1

The vertical asymptote is `x=root(3)6`
The horizontal asymptote is `y=0`
No oblique asymptote

Explanation:

Let `f(x)=(x+5)/(6-x^3)`

As you cannot divide by `0`, `x!=root(3)6`

The vertical asymptote is `x=root(3)6`

As the degree of the numerator is `<` than the degree of the denominator, there is no oblique asymptote

`lim_(x->-oo)f(x)=lim_(x->-oo)x/(-x^3)=lim_(x->-oo)-1/x^2=0^-`

`lim_(x->+oo)f(x)=lim_(x->+oo)x/(-x^3)=lim_(x->+oo)-1/x^2=0^-`

The horizontal asymptote is `y=0`
graph{(y-(x+5)/(6-x^3))(y)=0 [-20.27, 20.27, -10.14, 10.14]}