Given `h(x)=-4/(x-5),k(x)=2/(9-x),q(x)=5/(7x+1)` determine the horizontal and vertical asymptotes.
(1) All of these are in the form `f(x)=a/(bx+c)` . This is a rational function with constant numerator and linear denominator. In each case, the vertical asymptote will be `x=-c/b` , while the horizontal asymptote will be `y=0` .
For a rational function, there will be a vertical asymptote whenever the numerator and denominator have no common factors, and the denominator is zero.
For this type of rational function, note that `f(x)!=0` as the numerator is nonzero.
(2) `h(x)=(-4)/(x-5)` . There is a vertical asymptote at x=5, hor. asymptote at y=0.
(3) `k(x)=2/(9-x)` has a vertical asymptote when x=9, and a horizontal asymptote when y=0.
(4) `q(x)=5/(7x+1)` has a vertical asymptote at `x=-1/7` and a horizontal asymptote at y=0.
Please note that for `f(x)=a/(bx+c)+k` the horizontal asymptote is y=k, as the fraction is never zero.