Question

How do you find the vertical, horizontal or slant asymptotes for `f(x) = (3x + 5) / (7 - x)`?

Answer 1

vertical asymptote x = 7
horizontal asymptote y = -3

Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.

solve : 7 - x = 0 → x = 7 is the asymptote

Horizontal asymptotes occur as `lim_(xto+-oo) , f(x) to 0`

divide terms on numerator/denominator by x

`((3x)/x+5/x)/(7/x-x/x)=(3+5/x)/(7/x-1)`

as `xto+-oo , f(x)to(3+0)/(0-1)`

`rArry=-3" is the asymptote"`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1). Hence there are no slant asymptotes.
graph{(3x+5)/(7-x) [-40, 40, -20, 20]}