Please login to create an exam or a quiz.
Evaluate the limit \(\displaystyle \displaystyle\lim_{x \to -\infty}\left(15 x + \sqrt{225 x^{2} + 7 x + 8}\right)\)
Building the fraction by muliplying by the conjuate gives: \(\displaystyle \lim_{x\to-\infty}\frac{15 x + \sqrt{225 x^{2} + 7 x + 8}}{1}\cdot\left(\frac{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}\right)\) Simplifying gives \(\displaystyle \lim_{x \to -\infty} \frac{7 x + 8}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}\)factoring an \(\displaystyle x^2\) out of the radical in the denomiator and using the fact that \(\displaystyle \sqrt{x^2}=|x|\) gives \(\displaystyle \lim_{x \to -\infty} \frac{x \left(7 + \frac{8}{x}\right)}{- 15 x + \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} \left|{x}\right|}\) Using the fact that \(\displaystyle |x| =-x\) when \(\displaystyle x<0\) and reducing gives \(\displaystyle \lim_{x \to -\infty}\left(\frac{7 + \frac{8}{x}}{- \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} - 15}\right) = - \frac{7}{30}\)
\begin{question}Evaluate the limit $\displaystyle\lim_{x \to -\infty}\left(15 x + \sqrt{225 x^{2} + 7 x + 8}\right)$ \soln{9cm}{Building the fraction by muliplying by the conjuate gives: $\lim_{x\to-\infty}\frac{15 x + \sqrt{225 x^{2} + 7 x + 8}}{1}\cdot\left(\frac{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}\right)$ Simplifying gives $\lim_{x \to -\infty} \frac{7 x + 8}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}$factoring an $x^2$ out of the radical in the denomiator and using the fact that $\sqrt{x^2}=|x|$ gives $\lim_{x \to -\infty} \frac{x \left(7 + \frac{8}{x}\right)}{- 15 x + \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} \left|{x}\right|}$ Using the fact that $|x| =-x$ when $x<0$ and reducing gives $\lim_{x \to -\infty}\left(\frac{7 + \frac{8}{x}}{- \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} - 15}\right) = - \frac{7}{30}$ } \end{question}
\documentclass{article} \usepackage{tikz} \usepackage{amsmath} \usepackage[margin=2cm]{geometry} \usepackage{tcolorbox} \newcounter{ExamNumber} \newcounter{questioncount} \stepcounter{questioncount} \newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}} \renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}} \newif\ifShowSolution \newcommand{\soln}[2]{% \ifShowSolution% \noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else% \vspace{#1}% \fi% }% \newcommand{\hideifShowSolution}[1]{% \ifShowSolution% % \else% #1% \fi% }% \everymath{\displaystyle} \ShowSolutiontrue \begin{document}\begin{question}(10pts) The question goes here! \soln{9cm}{The solution goes here.} \end{question}\end{document}
<p> <p>Evaluate the limit <img class="equation_image" title=" \displaystyle \displaystyle\lim_{x \to -\infty}\left(15 x + \sqrt{225 x^{2} + 7 x + 8}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%5Cleft%2815%20x%20%2B%20%5Csqrt%7B225%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%7D%5Cright%29%20" alt="LaTeX: \displaystyle \displaystyle\lim_{x \to -\infty}\left(15 x + \sqrt{225 x^{2} + 7 x + 8}\right) " data-equation-content=" \displaystyle \displaystyle\lim_{x \to -\infty}\left(15 x + \sqrt{225 x^{2} + 7 x + 8}\right) " /> </p> </p>
<p> <p>Building the fraction by muliplying by the conjuate gives: <img class="equation_image" title=" \displaystyle \lim_{x\to-\infty}\frac{15 x + \sqrt{225 x^{2} + 7 x + 8}}{1}\cdot\left(\frac{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%5Cto-%5Cinfty%7D%5Cfrac%7B15%20x%20%2B%20%5Csqrt%7B225%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%7D%7D%7B1%7D%5Ccdot%5Cleft%28%5Cfrac%7B-%2015%20x%20%2B%20%5Csqrt%7B225%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%7D%7D%7B-%2015%20x%20%2B%20%5Csqrt%7B225%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%7D%7D%5Cright%29%20" alt="LaTeX: \displaystyle \lim_{x\to-\infty}\frac{15 x + \sqrt{225 x^{2} + 7 x + 8}}{1}\cdot\left(\frac{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}\right) " data-equation-content=" \displaystyle \lim_{x\to-\infty}\frac{15 x + \sqrt{225 x^{2} + 7 x + 8}}{1}\cdot\left(\frac{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}}\right) " /> Simplifying gives <img class="equation_image" title=" \displaystyle \lim_{x \to -\infty} \frac{7 x + 8}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20%5Cfrac%7B7%20x%20%2B%208%7D%7B-%2015%20x%20%2B%20%5Csqrt%7B225%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%7D%7D%20" alt="LaTeX: \displaystyle \lim_{x \to -\infty} \frac{7 x + 8}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}} " data-equation-content=" \displaystyle \lim_{x \to -\infty} \frac{7 x + 8}{- 15 x + \sqrt{225 x^{2} + 7 x + 8}} " /> factoring an <img class="equation_image" title=" \displaystyle x^2 " src="/equation_images/%20%5Cdisplaystyle%20x%5E2%20" alt="LaTeX: \displaystyle x^2 " data-equation-content=" \displaystyle x^2 " /> out of the radical in the denomiator and using the fact that <img class="equation_image" title=" \displaystyle \sqrt{x^2}=|x| " src="/equation_images/%20%5Cdisplaystyle%20%5Csqrt%7Bx%5E2%7D%3D%7Cx%7C%20" alt="LaTeX: \displaystyle \sqrt{x^2}=|x| " data-equation-content=" \displaystyle \sqrt{x^2}=|x| " /> gives <img class="equation_image" title=" \displaystyle \lim_{x \to -\infty} \frac{x \left(7 + \frac{8}{x}\right)}{- 15 x + \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} \left|{x}\right|} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20%5Cfrac%7Bx%20%5Cleft%287%20%2B%20%5Cfrac%7B8%7D%7Bx%7D%5Cright%29%7D%7B-%2015%20x%20%2B%20%5Csqrt%7B225%20%2B%20%5Cfrac%7B7%7D%7Bx%7D%20%2B%20%5Cfrac%7B8%7D%7Bx%5E%7B2%7D%7D%7D%20%5Cleft%7C%7Bx%7D%5Cright%7C%7D%20" alt="LaTeX: \displaystyle \lim_{x \to -\infty} \frac{x \left(7 + \frac{8}{x}\right)}{- 15 x + \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} \left|{x}\right|} " data-equation-content=" \displaystyle \lim_{x \to -\infty} \frac{x \left(7 + \frac{8}{x}\right)}{- 15 x + \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} \left|{x}\right|} " /> Using the fact that <img class="equation_image" title=" \displaystyle |x| =-x " src="/equation_images/%20%5Cdisplaystyle%20%7Cx%7C%20%3D-x%20" alt="LaTeX: \displaystyle |x| =-x " data-equation-content=" \displaystyle |x| =-x " /> when <img class="equation_image" title=" \displaystyle x<0 " src="/equation_images/%20%5Cdisplaystyle%20x%3C0%20" alt="LaTeX: \displaystyle x<0 " data-equation-content=" \displaystyle x<0 " /> and reducing gives <img class="equation_image" title=" \displaystyle \lim_{x \to -\infty}\left(\frac{7 + \frac{8}{x}}{- \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} - 15}\right) = - \frac{7}{30} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%5Cleft%28%5Cfrac%7B7%20%2B%20%5Cfrac%7B8%7D%7Bx%7D%7D%7B-%20%5Csqrt%7B225%20%2B%20%5Cfrac%7B7%7D%7Bx%7D%20%2B%20%5Cfrac%7B8%7D%7Bx%5E%7B2%7D%7D%7D%20-%2015%7D%5Cright%29%20%3D%20-%20%5Cfrac%7B7%7D%7B30%7D%20" alt="LaTeX: \displaystyle \lim_{x \to -\infty}\left(\frac{7 + \frac{8}{x}}{- \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} - 15}\right) = - \frac{7}{30} " data-equation-content=" \displaystyle \lim_{x \to -\infty}\left(\frac{7 + \frac{8}{x}}{- \sqrt{225 + \frac{7}{x} + \frac{8}{x^{2}}} - 15}\right) = - \frac{7}{30} " /> </p> </p>