Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function. R(x)= (5x^2-19x-4)/ 2x^2-7x-4
Find the vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
The vertical asymptotes occur when the denominator of the rational function is equal to zero.
To find the vertical asymptotes, set the denominator equal to zero and solve for x:
2x^2 - 7x - 4 = 0
Using the quadratic formula, we get:
x = (7 ± √(7^2 + 4*2*4)) / 4
x = (7 ± √(49 + 32)) / 4
x = (7 ± √81) / 4
x = (7 ± 9) / 4
This gives us two possible vertical asymptotes:
x = (7 + 9) / 4 = 4
x = (7 - 9) / 4 = -1/2
Therefore, the vertical asymptotes for the rational function R(x) are x = 4 and x = -1/2.