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calculus:resources:calculus_flipped_resources:applications:3.4_horizontal_asymptotes_tex

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.

\begin{document} \begin{frame} Find the following limits, if they exist. \vskip 5pt \begin{itemize} \item[\bf a)] $\dlim_{x\to\infty}\dfrac{7x^2 - x + 1}{3x^2 + 5x - 5}$ and $\dlim_{x\to-\infty}\dfrac{7x^2 - x + 1}{3x^2 + 5x - 5}$. \vskip 30pt \item[\bf b)] $\dlim_{x\to\infty}\dfrac{8x - 9}{2x + 4}$ and $\dlim_{x\to-\infty}\dfrac{8x - 9}{2x + 4}$. \vskip 30pt \item[\bf c)] $\dlim_{x\to\infty}\dfrac{x - 8}{x^2 + 7}$ and $\dlim_{x\to-\infty}\dfrac{x - 8}{x^2 + 7}$. \end{itemize} \end{frame} \begin{frame} Find the following limits, if they exist. \vskip 5pt \begin{itemize} \item[\bf d)] $\dlim_{x\to\infty}\dfrac{\sqrt{4x^6-x}}{x^3+3}$ and $\dlim_{x\to-\infty}\dfrac{\sqrt{4x^6-x}}{x^3+3}$. \vskip 20pt \item[\bf e)] $\dlim_{x\to\infty}(\sqrt{25x^2+x}-5x)$ and $\dlim_{x\to-\infty}(\sqrt{25x^2+x}-5x)$. \vskip 20pt \pause \item[\bf f)] $\dlim_{x\to-\infty}(x+\sqrt{x^2+2x})$ \vskip 10pt \item[\bf g)] $\dlim_{x\to\infty} 6\cos(x)$ \vskip 10pt \item[\bf h)] $\dlim_{x\to\infty}\frac{x^4 - 3x^2 + x}{x^3 - x + 3}$ \end{itemize} \end{frame} \begin{frame} Find the horizontal and vertical asymptotes of each curve. \begin{enumerate}[a)] \item $$y=\frac{8x + 3}{x - 4}$$ \item $$y=\frac{x^2 + 1}{9x^2 - 80x - 9}$$ \item $$y=\frac{x^2 - x}{x^2 - 8x + 7}$$ \end{enumerate} \end{frame} \begin{frame} Let $P$ and $Q$ be polynomials with positive coefficients. \begin{enumerate}[a)] \item If the degree of $P$ is less than the degree of $Q$, what is $$\lim_{x\to\infty}\frac{P(x)}{Q(x)}?$$ \item If the degree of $P$ is greater than the degree of $Q$, what is $$\lim_{x\to\infty}\frac{P(x)}{Q(x)}?$$ \item If the degree of $P$ equals the degree of $Q$, what is $$\lim_{x\to\infty}\frac{P(x)}{Q(x)}?$$ \end{enumerate} \end{frame} \begin{frame} A tank contains 120 L of pure water. Brine that contains 25 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. \vskip 15pt \begin{enumerate}[a)] \item Find the concentration of salt after $t$ minutes (in grams per liter). \vskip 15pt \item As $t$ approaches infinity, what does the concentration approach? \end{enumerate} \end{frame} \begin{frame} Find $$\lim_{x\to\infty}(\sqrt{x^2+cx}-\sqrt{x^2+dx}).$$ (Here $c$ and $d$ represent arbitrary real numbers.) \vskip 25pt Find $$\lim_{x\to -\infty}(x^2+x^3).$$ \end{frame} \end{document}