Question

characteristics of rational functions
determine the horizontal asymptote for each of the following functions. if it doesnt exist, write dne.
rational function horizontal asymptote
$f(x)=\frac{-5}{7 - 7x^{2}}$
$g(x)=\frac{24x - 8}{6x - 9}$
$f(x)=\frac{x + 5}{3x^{2}-1}$
$p(t)=\frac{t^{2}}{-6t - 4}$
question help: video message instructor

Explanation:

Step1: Recall horizontal - asymptote rules

For a rational function $y = \frac{f(x)}{g(x)}=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, if $n\lt m$, the horizontal asymptote is $y = 0$; if $n=m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$; if $n\gt m$, there is no horizontal asymptote (DNE).

Step2: Analyze $f(x)=\frac{-5}{7 - 7x^2}$

Here, $n = 0$ (degree of the numerator) and $m = 2$ (degree of the denominator). Since $n\lt m$, the horizontal asymptote is $y = 0$.

Step3: Analyze $g(x)=\frac{24x - 8}{6x - 9}$

The degree of the numerator $n = 1$ and the degree of the denominator $m = 1$. Then $a_n=24$ and $b_m = 6$. So, $y=\frac{24}{6}=4$ is the horizontal asymptote.

Step4: Analyze $f(x)=\frac{x + 5}{3x^2-1}$

The degree of the numerator $n = 1$ and the degree of the denominator $m = 2$. Since $n\lt m$, the horizontal asymptote is $y = 0$.

Step5: Analyze $p(t)=\frac{t^2}{-6t - 4}$

The degree of the numerator $n = 2$ and the degree of the denominator $m = 1$. Since $n\gt m$, there is no horizontal asymptote, so we write DNE.

Answer:

For $f(x)=\frac{-5}{7 - 7x^2}$, the horizontal asymptote is $y = 0$.
For $g(x)=\frac{24x - 8}{6x - 9}$, the horizontal asymptote is $y = 4$.
For $f(x)=\frac{x + 5}{3x^2-1}$, the horizontal asymptote is $y = 0$.
For $p(t)=\frac{t^2}{-6t - 4}$, the horizontal asymptote is DNE.