How do you find the slant asymptote of f(x)=( 21 x^2 ) / ( 3 x + 7)?

Jul 6, 2016

$y = 7 x$

Explanation:

$f \left(x\right) = \frac{21 {x}^{2}}{3 x + 7}$

you ask specifically about slants wcih means we consider $x \to \pm \infty$

${\lim}_{x \to \pm \infty} \frac{21 {x}^{2}}{3 x + 7}$

$= {\lim}_{x \to \pm \infty} \frac{21 x}{3 + \frac{7}{x}}$

and ${\lim}_{x \to \pm \infty} \left(3 + \frac{7}{x}\right) = 3$

so we can identify $y = \frac{21}{3} x = 7 x$ as a slant asymptote