Answer
The graph of the rational function, $f\left( x \right)=\frac{3{{x}^{2}}+1}{{{x}^{2}}+2}$ , will be:
Work Step by Step
Take the rational function;
$f\left( x \right)=\frac{3{{x}^{2}}+1}{{{x}^{2}}+2}$.
The equation of the horizontal line is the ratio of the coefficient of the higher degree term of the numerator and denominator.
So,
$\begin{align}
& y=\frac{3}{1} \\
& =3
\end{align}$
The conditions that are given are $x\to -\infty $ and $f\left( x \right)\to 3$ ,
Then,
$\begin{align}
& \underset{x\to -\infty }{\mathop{\lim }}\,f\left( x \right)=\underset{x\to -\infty }{\mathop{\lim }}\,\frac{3{{x}^{2}}+1}{{{x}^{2}}+2} \\
& =\underset{x\to -\infty }{\mathop{\lim }}\,\frac{{{x}^{2}}\left( 3+\frac{1}{{{x}^{2}}} \right)}{{{x}^{2}}\left( 1+\frac{2}{{{x}^{2}}} \right)} \\
& =\underset{x\to -\infty }{\mathop{\lim }}\,\frac{\left( 3+\frac{1}{{{x}^{2}}} \right)}{\left( 1+\frac{2}{{{x}^{2}}} \right)} \\
& =3
\end{align}$
Which satisfies the given condition. Hence, it is the required function.
