Question

for each function, determine the long run behavior (end behavior):
$f(x)=\frac{x^{2}+1}{x^{3}+2}$ has select an answer
$f(x)=\frac{x^{3}+1}{x^{2}+2}$ has select an answer
$f(x)=\frac{x^{2}+1}{x^{2}+2}$ has select an answer
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select an answer
no horizontal asymptote
a horizontal asymptote at y=0
a horizontal asymptote at y=1

Explanation:

Step1: Recall horizontal - asymptote rules

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

Step2: Analyze $f(x)=\frac{x^{2}+1}{x^{3}+2}$

Here $n = 2$ (degree of numerator) and $m = 3$ (degree of denominator). Since $n\lt m$, the horizontal asymptote is $y = 0$.

Step3: Analyze $f(x)=\frac{x^{3}+1}{x^{2}+2}$

Here $n = 3$ and $m = 2$. Since $n\gt m$, there is no horizontal asymptote.

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

Here $n = 2$ and $m = 2$, and $a_n = 1$, $b_m=1$. So $y=\frac{a_n}{b_m}=1$ is the horizontal asymptote.

Answer:

1. $f(x)=\frac{x^{2}+1}{x^{3}+2}$ has a Horizontal asymptote at $y = 0$
2. $f(x)=\frac{x^{3}+1}{x^{2}+2}$ has No horizontal asymptote
3. $f(x)=\frac{x^{2}+1}{x^{2}+2}$ has a Horizontal asymptote at $y = 1$