Find the vertical horizontal and oblique asymptotes if any f

Find the vertical, horizontal, and oblique asymptotes, if any, for the given rational function. R(x)= 12x2+19x21/3x+7

Solution

R(x)= (12x2+19x21)/(3x+7)

Vertical asymtote: The vertical asymptotes (and any restrictions on the domain) come from the zeroes of the denominator, so I\'ll set the denominator equal to zero and solve.

R(x) = (3x+7)(4x-3)/(3x +7) = 4x-3

There are no vertical asymptotes when the domain of x is all real numbers or when the term that creates the domain restriction cancels.

Horizontal asymtote: R(x)= (12x2+19x21)/(3x+7)

Now numerator can be factorised and written as : R(x) = (3x+7)(4x-3)/(3x +7) = 4x-3

Since the degree of the numerator is one greater than the degree of the denominator, I\'ll have a slant asymptote (not a horizontal one),

So, slant or oblique asymtote :y = 4x-3

Vertical asymtote : none

Horizontal asymtote : none

Oblique asymtote: y = 4x -3