# Chapter 2 - Section 2.6 - Rational Functions and Their Graphs - Exercise Set - Page 399: 40

$g\left( x \right)=5\text{ or }y=5$ is the equation of the horizontal asymptote.

#### Work Step by Step

Write $g\left( x \right)$ in the form $\frac{p\left( x \right)}{q\left( x \right)}$. \begin{align} & g\left( x \right)=\frac{15{{x}^{2}}}{3{{x}^{2}}+1} \\ & =\frac{p\left( x \right)}{q\left( x \right)} \end{align} To find the horizontal asymptotes, divide the numerator and denominator by the highest degree of x, which is 2. \begin{align} & g\left( x \right)=\frac{15{{x}^{2}}}{3{{x}^{2}}+1} \\ & =\frac{\frac{15{{x}^{2}}}{{{x}^{2}}}}{\frac{3{{x}^{2}}}{{{x}^{2}}}+\frac{1}{{{x}^{2}}}} \end{align} Now, when x tends to infinity, $\frac{1}{x}$ tends to zero. \begin{align} & g\left( x \right)=\frac{15{{x}^{2}}}{3{{x}^{2}}+1} \\ & =\frac{\frac{15{{x}^{2}}}{{{x}^{2}}}}{\frac{3{{x}^{2}}}{{{x}^{2}}}+\frac{1}{{{x}^{2}}}} \\ & =\frac{15}{3} \\ & =5 \end{align} Therefore, $g\left( x \right)=5\text{ or }y=5$ is the equation of the horizontal asymptote.

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