Question

How do you find the vertical, horizontal or slant asymptotes for `f(x) = (3x)/(x-8)`?

Answer 1

vertical asymptote x = 8
horizontal asymptote y = 3

Explanation:

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

solve : x - 8 = 0 → x = 8 is the asymptote

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

divide all terms on numerator/denominator by x

`((3x)/x)/(x/x - 8/x) = 3/(1 - 8/x)`

As x`to+-oo , 8/x to 0`

`rArr y = 3/1 = 3 " is the asymptote "`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes.

Here is the graph of f(x)
graph{(3x)/(x-8) [-10, 10, -5, 5]}