R(x)=(x^3-27)/(x^2-4x+3) find vertical, horizontal, and obliques asymptotes
To find the vertical asymptotes of the rational function R(x), we need to identify the values of x for which the denominator becomes zero (since division by zero is undefined).
First, factorize the denominator:
x^2 - 4x + 3 = (x - 1)(x - 3)
Now, set the denominator equal to zero and solve for x:
x - 1 = 0 --> x = 1
x - 3 = 0 --> x = 3
Therefore, we have two vertical asymptotes: x = 1 and x = 3.
To determine the horizontal asymptote, we compare the degrees of the numerator and the denominator.
In this case, the degree of the numerator (highest power of x) is 3, while the degree of the denominator is 2. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Oblique (slant) asymptotes occur when the degree of the numerator is exactly one higher than the degree of the denominator. In this case, since the degree of the numerator is greater, there are no oblique asymptotes either.
Hence, the vertical asymptotes of R(x) are x = 1 and x = 3, while there are no horizontal or oblique asymptotes.