Question

How do you find the vertical, horizontal or slant asymptotes for `(7x+7)/(x-2)`?

Answer 1

vertical asymptote x = 2
horizontal asymptote y = 7

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: x - 2 = 0 → x = 2 is the asymptote

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

divide terms on numerator/denominator by x

`((7x)/x+7/x)/(x/x-2/x)=(7+7/x)/(1-2/x)`

as `x to +- oo , y to(7+0)/(1-0)`

`rArr y=7" is the asymptote "`

Slant asymptotes occur when the degree of the numerator > degree of denominator. This is not the case here (both degree 1) hence there are no slant asymptotes.
graph{(7x+7)/(x-2) [-40, 40, -20, 20]}