Chapter 3 - Applications of Differentiation - 3.5 Exercises - Page 204: 81

$$\eqalign{ & {\text{Vertical asymptotes: none}} \cr & {\text{Horizontal asymptotes at }}y = \sin \left( 1 \right) \cr & {\text{Relative maximum at }}\left( {\frac{{2\pi }}{{\pi - 2}},1} \right) \cr & {\text{Domain}}:\left( {3,\infty } \right) \cr} $$

$$\eqalign{ & g\left( x \right) = \sin \left( {\frac{x}{{x - 2}}} \right),{\text{ }}x > 3 \cr & {\text{The domain is }}D:\left( {3,\infty } \right) \cr & {\text{*Differentiating}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {\sin \left( {\frac{x}{{x - 2}}} \right)} \right] \cr & f'\left( x \right) = \cos \left( {\frac{x}{{x - 2}}} \right)\frac{d}{{dx}}\left[ {\frac{x}{{x - 2}}} \right] \cr & f'\left( x \right) = \cos \left( {\frac{x}{{x - 2}}} \right)\left( {\frac{{x - 2 - x}}{{{{\left( {x - 2} \right)}^2}}}} \right) \cr & f'\left( x \right) = - \frac{2}{{{{\left( {x - 2} \right)}^2}}}\cos \left( {\frac{x}{{x - 2}}} \right),{\text{ }}x > 3 \cr & {\text{Set }}f'\left( x \right) = 0 \cr & - \frac{2}{{{{\left( {x - 2} \right)}^2}}}\cos \left( {\frac{x}{{x - 2}}} \right) = 0 \cr & \frac{x}{{x - 2}} = \frac{\pi }{2} \cr & {\text{Solving we obtain}} \cr & x = \frac{{2\pi }}{{\pi - 2}},{\text{ then by the first derivative test}}{\text{.}} \cr & {\text{ On the interval }}\left( {3,\frac{{2\pi }}{{\pi - 2}}} \right),{\text{ }}f'\left( x \right) > 0,{\text{ Increasing}} \cr & {\text{ On the interval }}\left( {\frac{{2\pi }}{{\pi - 2}},\infty } \right),{\text{ }}f'\left( x \right) < 0,{\text{ Decreasing}} \cr & {\text{There is a relative maximum at }}x = \frac{{2\pi }}{{\pi - 2}} \cr & g\left( {\frac{{2\pi }}{{\pi - 2}}} \right) = \sin \left( {\frac{{\frac{{2\pi }}{{\pi - 2}}}}{{\frac{{2\pi }}{{\pi - 2}} - 2}}} \right) = 1 \cr & {\text{Relative maximum at }}\left( {\frac{{2\pi }}{{\pi - 2}},1} \right) \cr & \cr & {\text{*There are no vertical asymptotes because the denominator}} \cr & {\text{is never 0}}{\text{.}} \cr & *{\text{Find the horizontal asymptotes}} \cr & \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \sin \left( {\frac{x}{{x - 2}}} \right) = \sin \left( 1 \right) \cr & {\text{Horizontal asymptotes at }}y = \sin \left( 1 \right) \cr & \cr & {\text{Summary}} \cr & {\text{Vertical asymptotes: none}} \cr & {\text{Horizontal asymptotes at }}y = \sin \left( 1 \right) \cr & {\text{Relative maximum at }}\left( {\frac{{2\pi }}{{\pi - 2}},1} \right) \cr & {\text{Domain}}:\left( {3,\infty } \right) \cr & \cr & {\text{Graph}} \cr} $$