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
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asked 2021-06-05
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?

Answers (1)

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
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