Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `f(x)=( 21 x^2 ) / ( 3 x + 7)`?

Answer 1

The vertical asymptote is `x=-7/3`
The oblique asymptote is `y=7x`
There is no horizontal asymptote

Explanation:

As you cannot divide by `0`, so `x=-7/3` is a vertical asymptote.

As the degree of the numerator `>` the degre of the denominator,
therefore, we expect an oblique asymptote.

Let's do a long division
`21x^2` `color(white)(aaaaaaaaa)` `∣` `3x+7`
`21x^2+49x` `color(white)(aaaa)` `∣` `7x`
`color(white)(aa)` `0-49x`

`:. (21x^2)/(3x+7)=7x-(49x)/(3x+7)`
So `y=7x` is an oblique asymptote
`lim_(x->+-oo)f(x)=lim_(x->+-oo)7x=+-oo`
So there is no horizontal asymptote
graph{(y-((21x^2)/(3x+7)))(y-7x)=0 [-277.3, 263.8, -233.8, 37]}