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

1 Answer
Sep 16, 2016

Answer:

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]}