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.

1 answer

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.