How do you use limits to find a horizontal asymptote?

Mar 17, 2018

Explanation:

We find limit of the function $f \left(x\right)$ as $x \to \infty$ i.e. $y = {\lim}_{x \to \infty} f \left(x\right)$. An example is shown below.

Let the function be $f \left(x\right) = \frac{a {x}^{3} + b {x}^{2} + c x + d}{p {x}^{3} + q {x}^{2} + r x + s}$,

then ${\lim}_{x \to \infty} \frac{a {x}^{3} + b {x}^{2} + c x + d}{p {x}^{3} + q {x}^{2} + r x + s}$.

Now dividing numerator and denominator by ${x}^{3}$, we get

${\lim}_{x \to \infty} \frac{a + \frac{b}{x} + \frac{c}{x} ^ 2 + \frac{d}{x} ^ 3}{p + \frac{q}{x} + \frac{r}{x} ^ 2 + \frac{s}{x} ^ 3}$

= $\frac{a}{p}$

and hence horizontal asymptote is $y = \frac{a}{p}$