Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `f(x)=x/(x^2+4)`?

Answer 1

The horizontal asymptote is `y=0`
No vertical asymptote
No oblique asymptote

Explanation:

The domain of `f(x)` is `D_f(x)=RR`

The denominator is `>0 AA x in RR`

There are no vertical asymptotes.

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

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

The horizontal asymptote is `y=0`

graph{(y-x/(x^2+4))(y)=0 [-5.55, 5.55, -2.773, 2.776]}