Question

How do you find the vertical, horizontal or slant asymptotes for `f(x) = (3x) /( x+4)`?

Answer 1

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

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+4=0rArrx=-4" is the asymptote"`

`"horizontal asymptotes occur as"`

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

`"divide terms on numerator/denominator by x"`

`f(x)=((3x)/x)/(x/x+4/x)=3/(1+4/x)`

`"as "xto+-oo,f(x)to3/(1+0)`

`y=3" is the asymptote"`

Slant asymptotes occur when the degree of the denominator is greater than the degree of the denominator. This is not the case here hence there are no slant asymptotes.
graph{(3x)/(x+4) [-20, 20, -10, 10]}