How do you find the vertical, horizontal and slant asymptotes of: #f(x)= (4x^2+ 4x-24)/(x^4- 2x^3 - 9x^2+ 18x)#?

Answer 1

#f(x)# has vertical asymptotes #x=0# and #x=3#, a horizontal asymptote #y=0# and holes (removable singularities) at #x=2# and #x=-3#.

The numerator factors like this:

#4x^2+4x-24 = 4(x^2+x-6) = 4(x+3)(x-2)#

The denominator factors like this:

#x^4-2x^3-9x^2+18x#
#= x((x^3-2x^2)-(9x-18))#
#= x(x^2(x-2)-9(x-2))#
#= x(x^2-9)(x-2)#
#= x(x-3)(x+3)(x-2)#
Note the common factors #(x-2)# and #(x+3)# in the numerator and denominator, so #f(x)# can be simplified, noting that #x=2# and #x = -3# are excluded from the domain:
#f(x) = (4x^2+4x-24)/(x^4-2x^3-9x^2+18x)#
#= (4color(red)(cancel(color(black)((x+3))))color(red)(cancel(color(black)((x-2)))))/(x(x-3)color(red)(cancel(color(black)((x+3))))color(red)(cancel(color(black)((x-2)))))#
#= (4)/(x(x-3))#
So when #x=0# or #x=3#, the denominator of #f(x)# is #0# and the numerator is non-zero, resulting in a vertical asymptote.
As #x->+-oo# the denominator #->+oo# resulting in a horizontal asymptote #y=0#.
#f(x)# has removable singularities (holes) at #x=2# and #x=-3#

graph{(4x^2+4x-24)/(x^4-2x^3-9x^2+18x) [-10, 10, -5, 5]}

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Answer 2

To find the vertical asymptotes, set the denominator equal to zero and solve for ( x ). For horizontal asymptotes, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is ( y = 0 ). If the degrees are equal, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote. If the degree of the numerator is greater, there is no horizontal asymptote. To find slant asymptotes, perform polynomial long division and the slant asymptote is the quotient.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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