Question

# Consider the following rational functions: r(x)=(2x−1)/((x^2)−x−2) s(x)=((x^3)+27)/((x^2)+4) t(x)=((x^3)−9x)/(x+2) u(x)=((x^2)+x−6)/((x^2)−25) w(x)=((x^3)+(6x^2)+9x)/(x+3) Which of these rational functions has a horizontal asymptote?

Rational functions
Consider the following rational functions: $$\displaystyle{r}{\left({x}\right)}=\frac{{{2}{x}−{1}}}{{{\left({x}^{{2}}\right)}−{x}−{2}}}$$
$$\displaystyle{s}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{27}}}{{{\left({x}^{{2}}\right)}+{4}}}$$
$$\displaystyle{t}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}−{9}{x}}}{{{x}+{2}}}$$
$$\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}−{6}}}{{{\left({x}^{{2}}\right)}−{25}}}$$
$$\displaystyle{w}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{\left({6}{x}^{{2}}\right)}+{9}{x}}}{{{x}+{3}}}$$
Which of these rational functions has a horizontal asymptote?

2021-06-06
$$\displaystyle{r}{\left({x}\right)}=\frac{{{2}{x}−{1}}}{{{\left({x}^{{2}}\right)}−{x}−{2}}}=\frac{{{2}{x}-{1}}}{{{\left({x}-{2}\right)}{\left({x}+{1}\right)}}}$$
$$\displaystyle{s}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{27}}}{{{\left({x}^{{2}}\right)}+{4}}}=\frac{{{\left({x}+{3}\right)}{\left({x}+{9}+{3}{x}\right)}}}{{{\left({x}^{{2}}\right)}+{4}}}$$
$$\displaystyle{t}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}−{9}{x}}}{{{x}+{2}}}=\frac{{{x}{\left({x}-{3}\right)}{\left({x}+{3}\right)}}}{{{x}+{2}}}$$
$$\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}−{6}}}{{{\left({x}^{{2}}\right)}−{25}}}=\frac{{{\left({x}-{2}\right)}{\left({x}+{3}\right)}}}{{{\left({x}-{5}\right)}{\left({x}+{5}\right)}}}$$
$$\displaystyle{w}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{\left({6}{x}^{{2}}\right)}+{9}{x}}}{{{x}+{3}}}={x}{\left({x}+{3}\right)}$$
r,t and u have horizontal asymptotes