Let F(x) = -4x^2-4x+48/2x^2-10x+12

Question

A. Finds lim f(x) x->-infinity and state which shortcut rule was used

B. Find the equation of any horizontal asymptote

C. Find lim f(x) x->2

D. Find lim f(x) x->3

E. Find the coordinates of any removable discontinuities

F. Find the equation of any vertical asymptote

Answer 1

Factor out -4 in numerator and 2 in the denominator, factor

-4(x²+x-12)/(2[x²-5x+6)]

-4(x+4)(x-3)/2[(x-2)(x-3)] removable discontinuity x-3

-2(x+4)/(x-2)

as x→∞ -1

Horizontal Asymptote x=-2

x→2 -∞

x→3 undefined, removable discontinuity

Vertical Asymptote x=2

Answer 2

A. LIM (x->∞) F(x) = -4/2 = -2. This is because we compare the highest powers to find the limit

B. The horizontal asymptote would therefore be y=-2

C. The limit does not exist when f(x) ->2, since the left and right hand limits do not agree

D. Again, this limit does not exist due to the left and right hand limits not agreeing

E. Removable discontinuities occur at x=3 and 2. When you factor these points are "holes" meaning undefined.

F. The vertical asymptote would therefore be at x=2. Because when you factor out and set x-2, the denominator becomes 0 and the function becomes undefined.