Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function. R(x)= (x^3-8)/ x^2-3x+2
Find the oblique asymptotes. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
To find the vertical asymptotes, we need to find the values of x that make the denominator equal to zero.
x^2 - 3x + 2 = 0
(x - 1)(x - 2) = 0
Therefore, x = 1 and x = 2 are the vertical asymptotes.
To find the horizontal asymptote, we compare the degrees of the numerator and denominator.
Since the degree of the numerator (3) is greater than the degree of the denominator (2), there is no horizontal asymptote.
To find the oblique asymptote, we divide the numerator by the denominator using long division or synthetic division.
(x^3 - 8) / (x^2 - 3x + 2) = x + 3 + 1/(x - 2)
Therefore, the oblique asymptote is y = x + 3.
Final answer:
Vertical asymptotes: x = 1, x = 2
Horizontal asymptote: None
Oblique asymptote: y = x + 3