Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
[tex]\lim\limits_{x\to0}\frac{\sqrt{cos2x}-\sqrt[3]{cosx}}{arctanx^2}=\left[\frac{0}{0}\right]\\\\De\ L'Hospital's\ rule:\\\\\lim\limits_{x\to0}\frac{\sqrt{cos2x}-\sqrt[3]{cosx}}{arctanx^2}=\lim\limits_{x\to0}\frac{(\sqrt{cos2x}-\sqrt[3]{cosx})'}{(arctanx^2)'}=(*)\\-------------------------------\\(\sqrt{cos2x}-\sqrt[3]{cosx})'=\frac{-sin2x}{\sqrt{cos2x}}-\frac{-sinx}{3\sqrt[3]{cos^2x}}=\frac{-2sinxcosx\cdot3\sqrt[3]{cos^2x}+sinx\sqrt{cos2x}}{3\sqrt{cos2x}\cdot\sqrt[3]{cos^2x}}\\\\(arctanx^2)'=\frac{2x}{x^2+1}[/tex]
[tex]-------------------------------\\(*)=\lim\limits_{x\to0}\frac{-6sinxcosx\sqrt[3]{cos^2x}+sinx\sqrt{cos2x}}{3\sqrt[3]{cos^2x}\cdot\sqrt{cos2x}}:\frac{2x}{x^4+1}\\\\=\lim\limits_{x\to0}\frac{-sinx(6cosx\sqrt[3]{cos^2x}-\sqrt{cos2x})}{3\sqrt[3]{cos^2x}\cdot\sqrt{cos2x}}\cdot\frac{x^4+1}{2x}\\\\=\lim\limits_{x\to0}\frac{-sinx}{x}\cdot\frac{(6cosx\sqrt[3]{cos^2x}-\sqrt{cos2x})(x^4+1)}{6}=-1\cdot\frac{(6\cdot1\sqrt[3]{1^2}-\sqrt1)(0+1)}{6}\\\\=-1\cdot\frac{5}{6}=-\frac{5}{6}[/tex]
[tex]-------------------------------\\(*)=\lim\limits_{x\to0}\frac{-6sinxcosx\sqrt[3]{cos^2x}+sinx\sqrt{cos2x}}{3\sqrt[3]{cos^2x}\cdot\sqrt{cos2x}}:\frac{2x}{x^4+1}\\\\=\lim\limits_{x\to0}\frac{-sinx(6cosx\sqrt[3]{cos^2x}-\sqrt{cos2x})}{3\sqrt[3]{cos^2x}\cdot\sqrt{cos2x}}\cdot\frac{x^4+1}{2x}\\\\=\lim\limits_{x\to0}\frac{-sinx}{x}\cdot\frac{(6cosx\sqrt[3]{cos^2x}-\sqrt{cos2x})(x^4+1)}{6}=-1\cdot\frac{(6\cdot1\sqrt[3]{1^2}-\sqrt1)(0+1)}{6}\\\\=-1\cdot\frac{5}{6}=-\frac{5}{6}[/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
