How do you find vertical, horizontal and oblique asymptotes for #R(x) = (6x^2 + x + 12)/(3x^2 - 5x - 2)#?

Answer 1

vertical asymptotes #x=-1/3,x=2#
horizontal asymptote y = 2

Since the denominator of R(x) is undefined, it cannot be zero. By solving for the denominator, one can determine the values that x cannot be. If the numerator of these values is non-zero, the values are vertical asymptotes.

solve: #3x^2-5x-2=0rArr(3x+1)(x-2)=0#
#rArrx=-1/3" and " x=2" are the asymptotes"#

As horizontal asymptotes arise,

#lim_(xto+-oo),R(x)toc" (a constant)"#
divide terms on numerator/denominator by the highest power of x, that is #x^2#
#((6x^2)/x^2+x/x^2+12/x^2)/((3x^2)/x^2-(5x)/x^2-2/x^2)=(6+1/x+12/x^2)/(3-5/x-2/x^2)#
as #xto+-oo,R(x)to(6+0+0)/(3-0-0)#
#rArry=2" is the asymptote"#

There are no oblique asymptotes in this case (both of degree 2), as oblique asymptotes arise when the degree of the numerator > degree of the denominator. graph{(6x^2+x+12)/(3x^2-5x-2) [-10, 10, -5, 5]}

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Answer 2

Vertical asymptotes: Set the denominator equal to zero and solve for x. The values of x obtained are the vertical asymptotes.

Horizontal asymptote: Compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, then 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.

Oblique asymptote: Perform polynomial long division to divide the numerator by the denominator. The quotient obtained represents the oblique asymptote.

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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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