Tutorial Exercise Find any horizontal and vertical asymptotes for the function. 4x y = - x - 5 Step 1 If lim f(x) = b or lim f(x) = b, then the function has a horizontal asymptote at y = b. Divide each term in the numerator and denominator by the highest power of x, then find the limit. x - 00 X -> -00 4 4x lim lim x -> too x- 5 x - + 00 1 0 1 5 0 5 = 4 Step 2 Therefore, there is a horizontal asymptote at the following. 4 X Notice that the asymptote is given by the ratio of the leading coefficients of the numerator and denominator. This is true when the degrees of the numerator and denominator are the same. Submit Skip (you cannot come back).Find any horizontal and vertical asymptotes for the function. (Enter your answers as a comma-separated list of equations. If an answer does not exist, enter DNE.) 3X y = . X - 6 horizontal asymptotes 3 vertical asymptotes 6 Need Help? Read It Watch It Master It 3. [0.75/1.36 Points] DETAILS PREVIOUS ANSWERS HARMATHAP12 10.5.004. A function and its graph are given. Use the graph to find each of the following, if they exist. Then confirm your results analytically. f ( x ) = - x 2 (x - 3) 2 15 10 5- -15 - 10 -5 5 10 15 a) vertical asymptotes (Enter your answers as a comma-separated list of equations.) 3 X (b) lim f(x) 1 c) lim f(x) (d) horizontal asymptotes (Enter your answers as a comma-separated list of equations.) 1Find any horizontal and vertical asymptotes for the function. (Enter your answers as a comma-separated list of equations. If an answer does not exist, enter DNE.) y = - x+ 3 x2 - 16 izontal asymptotes dne X vertical asymptotes dne X Need Help? Read It 5. [1.5/1.36 Points] DETAILS PREVIOUS ANSWERS HARMATHAP12 10.5.009. Find any horizontal and vertical asymptotes for the function. (Enter your answers as a comma-separated list of equations. If an answer does not exist, enter DNE.) y = _ 5x3 - 3 x-+1 horizontal asymptotes dne vertical asymptotes dne Need Help? Read It Watch It 6. [0/1.36 Points] DETAILS PREVIOUS ANSWERS HARMATHAP12 10.5.011. For the function, find any horizontal and vertical asymptotes. (Enter your answers as a comma-separated list of equations. If an answer does not exist, enter DNE.) f(x) = 4x + 12 X - 2 horizontal asymptotes 4 X vertical asymptotes 2 XFor the function, find any horizontal and vertical asymptotes. (Enter your answers as a comma-separated list of equations. If an answer does not exist, enter DNE.) f ( x ) = _ 21x2 (x + 1) 3 horizontal asymptotes 0 vertical asymptotes -1 X Use information from the first derivative to sketch the graph. =4 40 hot 40 20 20 20 20 -20 -10 10 X X LX -20 -10 10 20 -20 -10 10 20 -20 -10 10 20 -20 -20 20 -20 -40 -40 -40 O DO Need Help? Read It 8. [0.51/1.36 Points] DETAILS PREVIOUS ANSWERS HARMATHAP12 10.5.017. For the function, find any horizontal and vertical asymptotes. (Enter your answers as a comma-separated list of equations. If an answer does not exist, enter DNE.) 18x f ( x ) = - x2+ 5 horizontal asymptotes dne X vertical asymptotes 0A function and its first and second derivatives are given. Use these to find each of the following. (If an answer does not exist, enter DNE.) X y= ( x - 7)2 x+7 y' ( x - 7)3 2x + 28 y" = ( x - 7) 4 Find any horizontal and vertical asymptotes. (Enter your answers as a comma-separated list of equations.) horizontal asymptotes 0 X vertical asymptotes 3 X Find any critical points. ( x, y) = -3 Find any relative maxima and relative minima. relative maximum (x, y) = ane relative minimum ( x, y ) = -3, - 1 12 X Find any points of inflection. (x, y) =-6, - 27 XA function and its first and second derivatives are given. Use these to find each of the following. (If an answer does not exist, enter DNE.) f ( x ) = 18(x - 2) 2/3 x2 f' ( x ) = 24 (3 - X) x3(x - 2) 1/3 f" ( x ) = 3 8(7x2 - 42x + 54) x* ( x - 2) 4/3 Find any horizontal and vertical asymptotes. (Enter your answers as a comma-separated list of equations.) horizontal asymptotes vertical asymptotes Find any critical points. (x, y) = ( (smaller x-value) ( x, y ) = (larger x-value) Find any relative maxima and relative minima. relative maximum (x, y) = relative minimum ( x , y ) = Find any points of inflection. (Round the coordinates to two decimal places.) (x, y ) = (smaller x-value) ( x, y ) = (larger x-value)A function and its first and second derivatives are given. Use these to find each of the following. (If an answer does not exist, enter DNE.) (x) = 6x2/3 x +1 f' ( x ) =. 2(2 - X) x1/3(x + 1)2 f" ( x) = 4 4(2x2 - 8x - 1) 3x4/3(x + 1) 3 Find any horizontal and vertical asymptotes. (Enter your answers as a comma-separated list of equations.) horizontal asymptotes 0 X vertical asymptotes -1 X Find any critical points. (x, y) = ( 2,2(2) ( smaller x-value) X (x, y) = dne (larger x-value) Find any relative maxima and relative minima. relative maximum (x, y) = relative minimum ( x , y ) = a Find any points of inflection. (Round coordinates to two decimal places.) (x, y) = 0,0 (smaller x-value) ( x, y) = -1,00 (larger x-value) X