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# (a) Find lim f(x) and state which shortcut rule was used: Show all work.

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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:
x→-∞

(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.

Apr 4, 2020

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Translation:

Apr 4, 2020
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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

Apr 4, 2020