Question

What are the asymptotes of `f( x ) = \frac { 4x + 6} { x - 3}`?

Answer 1

`"vertical asymptote at "x=3`
`"horizontal asymptote at "y=4`

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)=((4x)/x+6/x)/(x/x-3/x)=(4+6/x)/(1-3/x)`

`"as "xto+-oo,f(x)to(4+0)/(1-0)`

`rArry=4" is the asymptote"`
graph{(4x+6)/(x-3) [-20, 20, -10, 10]}