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

Nov 8, 2016

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

#### Explanation:

As you cannot divide by $0$, so $x = - \frac{7}{3}$ 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
$21 {x}^{2}$$\textcolor{w h i t e}{a a a a a a a a a}$∣$3 x + 7$
$21 {x}^{2} + 49 x$$\textcolor{w h i t e}{a a a a}$∣$7 x$
$\textcolor{w h i t e}{a a}$$0 - 49 x$

$\therefore \frac{21 {x}^{2}}{3 x + 7} = 7 x - \frac{49 x}{3 x + 7}$
So $y = 7 x$ is an oblique asymptote
${\lim}_{x \to \pm \infty} f \left(x\right) = {\lim}_{x \to \pm \infty} 7 x = \pm \infty$
So there is no horizontal asymptote
graph{(y-((21x^2)/(3x+7)))(y-7x)=0 [-277.3, 263.8, -233.8, 37]}