I need someone to help me answer this difficult problem for me. This is important for me so I would appreciate if everything is clear for me to understand. Thanks!
(a) Find lim f(x) and state which shortcut rule was used:
(b) Find the equation of any horizontal asymptote:
(c) Find the y-intercept:
(d) Use algebra to find the equations of any vertical asymptotes and the coordinates of any holes (removable discontinuities).
(e) Find the x-intercept:
(f) Sketch a graph for f(x) showing the results from parts (a) through (e). Set up the graph for x-values from -8 to 8 with increments of 1 and y-values from -18 to 18 with increments of 2.
f(x) = (-4x2 -4x + 48) / (2x2 - 10x + 12)
a) Divide both the numerator and denominator by x2 ---> (-4 - 4/x + 48/x2) / (2 - 10/x + 12/x2)
as x approaches infinity, every term with an x in the denominator goes to zero; so ---> -4/2 = -2
b) Horizontal asymptote: y = -2
c) y-intercept: x = 0 ---> y = 4
d) Factor the numerator: -4x2 - 4x + 48 = -4(x2 + x - 12) = -4(x + 4)(x - 3)
Factor the denominator: 2x2 - 10x + 12 = 2(x2 - 5x + 6) = 2(x - 2)(x - 3)
Since both the numerator and denominator has a factor of 3, a removable discontinutity occurs at x= 3
A vertical asymptote occurs at x = 2.
e) x-intercept: y = 0 ---> x = -4