Question

find all horizontal asymptotes of the following function.
$f(x) = \frac{3x^2 + 27x}{2x^2 - 16x + 24}$
answer attempt 1 out of 2
two horizontal asymptotes
64 and

Explanation:

Step1: Divide numerator and denominator by \(x^{2}\)

$$\begin{align*} \lim_{x ightarrow\pm\infty}\frac{3x^{2}+27x}{2x^{2}-16x + 24}&=\lim_{x ightarrow\pm\infty}\frac{3+\frac{27}{x}}{2-\frac{16}{x}+\frac{24}{x^{2}}}\\ \end{align*}$$

Step2: Evaluate the limit as \(x

ightarrow\pm\infty\)
As \(x
ightarrow\pm\infty\), \(\frac{1}{x}
ightarrow0\) and \(\frac{1}{x^{2}}
ightarrow0\).
So, \(\lim_{x
ightarrow\pm\infty}\frac{3+\frac{27}{x}}{2-\frac{16}{x}+\frac{24}{x^{2}}}=\frac{3 + 0}{2-0 + 0}=\frac{3}{2}\)

Answer:

\(\frac{3}{2}\)