Question

How do you identify all asymptotes for `f(x)=(2x)/(x-3)`?

Answer 1

vertical asymptote at x = 3
horizontal asymptote at y = 2

Explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: `x-3=0rArrx=3" is the asymptote"`

Horizontal asymptotes occur as

`lim_(xto+-oo),f(x)toc" (a constant)"`

divide terms on numerator/denominator by x

`f(x)=((2x)/x)/(x/x-3/x)=2/(1-3/x)`

as `xto+-oo,f(x)to2/(1-0)`

`rArry=2" is the asymptote"`
graph{(2x)/(x-3) [-20, 20, -10, 10]}