Chapter 3 - Applications of Differentiation - 3.6 Exercises - Page 212: 10

$$\eqalign{ & {\text{No }}y{\text{ - intercept}} \cr & x{\text{ - intercept }}\left( {3,0} \right) \cr & {\text{No relative extrema}}{\text{.}} \cr & {\text{No inflection points}}{\text{.}} \cr & {\text{Vertical asymptotes at }}x = 0 \cr & {\text{Horizontal asymptote }}y = 1 \cr} $$

$$\eqalign{ & f\left( x \right) = \frac{{x - 3}}{x} \cr & y = 1 - \frac{3}{x} \cr & {\text{Find the }}y{\text{ intercept}}{\text{, let }}x = 0 \cr & y = 1 - \frac{3}{0} \cr & {\text{No }}y{\text{ - intercept}}{\text{.}} \cr & {\text{Find the }}x{\text{ intercept}}{\text{, let }}y = 0 \cr & 0 = 1 - \frac{3}{x} \cr & \frac{3}{x} = 1 \cr & x = 3 \cr & x{\text{ - intercept }}\left( {3,0} \right) \cr & \cr & *{\text{Find the extrema}} \cr & {\text{Differentiate}} \cr & y' = \frac{d}{{dx}}\left[ {1 - \frac{3}{x}} \right] \cr & y' = \frac{3}{{{x^2}}} \cr & {\text{Let }}y' = 0 \cr & \frac{3}{{{x^2}}},{\text{ there are no values at which }}y' = 0, \cr & {\text{No relative extrema}}{\text{.}} \cr & \cr & *{\text{Find the second derivative}} \cr & y'' = \frac{d}{{dx}}\left[ {\frac{3}{{{x^2}}}} \right] \cr & y'' = 3\left( { - 2{x^{ - 3}}} \right) \cr & y'' = - \frac{6}{{{x^3}}} \cr & {\text{Let }}y'' = 0 \cr & - \frac{6}{{{x^3}}} = 0 \cr & {\text{There are no values at which }}y'' = 0. \cr & {\text{No inflection points}}{\text{.}} \cr & \cr & {\text{*Calculate the asymptotes}} \cr & 1 - \frac{3}{x} \cr & \to x = 0 \cr & {\text{Vertical asymptotes at }}x = 0 \cr & \mathop {\lim }\limits_{x \to \infty } \left( {1 - \frac{3}{x}} \right) = 1 \cr & \mathop {\lim }\limits_{x \to - \infty } \left( {1 - \frac{3}{x}} \right) = 1 \cr & {\text{Horizontal asymptote }}y = 1 \cr & \cr & {\text{Graph}} \cr} $$