How do you find the Vertical, Horizontal, and Oblique Asymptote for #R(x)=(3x+5) \ (x-6)#?

Question

Elias Ackerman · Accepted Answer

Was the question intended to be for the function #R(x) = (3x+5)/(x-6)#?

The quotient is already reduced. (There are no common factors onf the numerator and the denominator). So we can find vertical asymptotes by solving denominator = #0#. The equation of the vertical asymptote is #x=6#. For values of #x# far from #0# (positive or negative), we have #R(x) = (cancel(x)(1+5/x))/(cancel(x)(1-6/x))# For #x# far from #0# (#x# with large absolute value), both #5/x# and #6/x# are close to #0#, so #R(x)# is close to #3/1 = 3#. The line #y=3# is a horizontal asymptote on both sides. The graph of a rational function cannot have both horizontal and obliques asymptotes. The graph of this function does have horizontal asymptotes, so it cannot have oblique asymptotes.

Elijah Allen · Answer

This function has no asymptotes.

The given function is a polynomial and polynomials do not have asymptotes.

Sophie Adams · Answer

undefined To find the vertical asymptote, set the denominator equal to zero and solve for x. For $ R(x) = \frac{3x + 5}{x - 6} $, the vertical asymptote occurs when $ x - 6 = 0 $, so $ x = 6 $ is the vertical asymptote.

To find the horizontal asymptote, 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, the horizontal asymptote is the ratio of the leading coefficients. In this case, since the degree of the numerator (1) is less than the degree of the denominator (1), the horizontal asymptote is y = 0.

To find the oblique asymptote, if the degree of the numerator is exactly one more than the degree of the denominator, perform polynomial long division. In this case, the degrees are equal, so there is no oblique asymptote.