for each function, determine the long run behavior.\n(x^2 + 1)/(x^2 + 2) has select an answer\n(x^3 +…

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 $n = m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$; if $n<m$, the horizontal asymptote is $y = 0$; if $n>m$, there is no horizontal asymptote.

Step2: Analyze $\frac{x^{2}+1}{x^{2}+2}$

Here $n = m=2$, $a_n = 1$, $b_m = 1$. So, $\lim_{x\rightarrow\pm\infty}\frac{x^{2}+1}{x^{2}+2}=\lim_{x\rightarrow\pm\infty}\frac{1+\frac{1}{x^{2}}}{1 + \frac{2}{x^{2}}}=1$. It has a horizontal asymptote at $y = 1$.

Step3: Analyze $\frac{x^{3}+1}{x^{2}+2}$

Here $n=3$, $m = 2$, $n>m$. So, $\lim_{x\rightarrow\pm\infty}\frac{x^{3}+1}{x^{2}+2}=\lim_{x\rightarrow\pm\infty}(x+\frac{1 - 2x}{x^{2}+2})=\pm\infty$. It has no horizontal asymptote.

Step4: Analyze $\frac{x^{2}+1}{x^{3}+2}$

Here $n = 2$, $m=3$, $n<m$. So, $\lim_{x\rightarrow\pm\infty}\frac{x^{2}+1}{x^{3}+2}=\lim_{x\rightarrow\pm\infty}\frac{\frac{1}{x}+\frac{1}{x^{3}}}{1+\frac{2}{x^{3}}}=0$. It has a horizontal asymptote at $y = 0$.

Answer:

1. a Horizontal asymptote at y = 1
2. No horizontal asymptote
3. a Horizontal asymptote at y = 0