find all horizontal and vertical asymptotes

Question

f(x) = 5x^2-21x+4/x^2-x-12

Answer 1

f(x) = (5x²-21x+4) / (x²-x-12) = (5x-1)(x-4) / (x-4)(x+3) = (5x-1) / (x+3)

 

Horizontal asymptote at the ratio of two highest degree terms: y = 5x/x = 5

Vertical asymptote where the denominator equals zero, at x = -3

 

However, there is a removable discontinuity at x=4 because the (x-4) in the denominator of the original equation cancels with the (x-4) in the numerator.  There is no asymptote at x = 4, but there is a hole as f(x) is not defined at x=4.