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calculus:resources:calculus_flipped_resources:applications:3.4_horizontal_asymptotes_tex [2014/09/06 17:13] jbrennan |
calculus:resources:calculus_flipped_resources:applications:3.4_horizontal_asymptotes_tex [2015/08/28 22:19] (current) nye Page moved from people:jbrennan:calculus_flipped_resources:applications:3.4_horizontal_asymptotes_tex to calculus:resources:calculus_flipped_resources:applications:3.4_horizontal_asymptotes_tex |
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+ | TeX code compiled with \documentclass{beamer} using the Amsterdam theme.\\ | ||
+ | <nowiki> | ||
+ | \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} | ||
+ | </nowiki> |