Single Variable Calculus, Chapter 4, 4.4, Section 4.4, Problem 56

By the end behavior of a function we mean the behavior of its values as $x \to \infty$ and as $x \to - \infty$.

a) Compare and describe the end behavior of the functions

$P(x) = 3x^5 - 5x^3 + 2x \qquad Q(x) = 3x^5$

By graphing both functions in the viewing rectangles $[-2, 2]$ by $[-2, 2]$ and $[-10, 10]$ by $[-10,000,10,000]$.














Based from the graph, as $x$ approaches $\infty$ both functions $P(x)$ and $Q(x)$ approaches positive infinity and as $x$ approaches $- \infty$ both functions $P(x)$ and $Q(x)$ approaches negative infinity.



b) Show that if $P$ and $Q$ have the same behavior if their ratio approaches 1 as $x \to \infty$




$
\begin{equation}
\begin{aligned}

\lim_{x \to \infty} \frac{P(x)}{Q(x)} =& \lim_{x \to \infty} \frac{3x^5 - 5x^3 + 2x}{3x^5}
\\
\\
\lim_{x \to \infty} \frac{P(x)}{Q(x)} =& \lim_{x \to \infty} \frac{\displaystyle \frac{3 \cancel{x^5}}{\cancel{x^5}} - \frac{5x^3}{x^3} + \frac{2x}{x^5} }{\displaystyle \frac{3 \cancel{x^5}}{\cancel{x^5}}}
\\
\\
\lim_{x \to \infty} \frac{P(x)}{Q(x)} =& \lim_{x \to \infty} \frac{\displaystyle 3 - \frac{5}{x^2} + \frac{2}{x^4}}{3}
\\
\\
\lim_{x \to \infty} \frac{P(x)}{Q(x)} =& \frac{\displaystyle 3 - \lim_{x \to \infty} \frac{5}{x^2} + \frac{2}{x^4} }{3}
\\
\\
\lim_{x \to \infty} \frac{P(x)}{Q(x)} =& \frac{3 - 0 + 0}{3}
\\
\\
\lim_{x \to \infty} \frac{P(x)}{Q(x)} =& \frac{3}{3}
\\
\\
\lim_{x \to \infty} \frac{P(x)}{Q(x)} =& 1

\end{aligned}
\end{equation}
$