Question

How to find the asymptotes of `f(x)=( 21 x^2 ) / ( 3 x + 7)`?

Answer 1

The vertical asymptote is `x=-7/3`
The oblique asymptote is `y=7x-49/3`
There are no horizontal asymptotes

Explanation:

As we cannot divide by `0`, the vertical asymptote is `x=-7/3`

The degree of the numerator is `>` than the degree of the numerator, so we expect a slant asymptote.

Let's do a long division

`color(white)(aaaa)` `21x^2` `color(white)(aaaaaaaaa)` `∣` `3x+7`
`color(white)(aaaa)` `21x^2+49x` `color(white)(aaaa)` `∣` `7x-49/3`
`color(white)(aaaaaaa)` `0-49x`
`color(white)(aaaaaaaaa)` `-49x-343/3`
`color(white)(aaaaaaaaaaa)` `-0+343/3`

`:. f(x)=(7x-49/3)+(343/3)/(3x+7)`
The oblique asymptote is `y=7x-49/3`
graph{(y-21x^2/(3x+7))(y-7x+49/3)=0 [-169, 168.7, -84.7, 84.5]}