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

Nov 12, 2016

The vertical asymptote is $x = - \frac{7}{3}$
The oblique asymptote is $y = 7 x - \frac{49}{3}$
There are no horizontal asymptotes

Explanation:

As we cannot divide by $0$, the vertical asymptote is $x = - \frac{7}{3}$

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

Let's do a long division

$\textcolor{w h i t e}{a a a a}$$21 {x}^{2}$$\textcolor{w h i t e}{a a a a a a a a a}$∣$3 x + 7$
$\textcolor{w h i t e}{a a a a}$$21 {x}^{2} + 49 x$$\textcolor{w h i t e}{a a a a}$∣$7 x - \frac{49}{3}$
$\textcolor{w h i t e}{a a a a a a a}$$0 - 49 x$
$\textcolor{w h i t e}{a a a a a a a a a}$$- 49 x - \frac{343}{3}$
$\textcolor{w h i t e}{a a a a a a a a a a a}$$- 0 + \frac{343}{3}$

$\therefore f \left(x\right) = \left(7 x - \frac{49}{3}\right) + \frac{\frac{343}{3}}{3 x + 7}$
The oblique asymptote is $y = 7 x - \frac{49}{3}$
graph{(y-21x^2/(3x+7))(y-7x+49/3)=0 [-169, 168.7, -84.7, 84.5]}