Question

How do you find the vertical, horizontal or slant asymptotes for `y=(13*x)/(x+34)`?

Answer 1

`"vertical asymptote at "x=-34`
`"horizontal asymptote at "y=13`

Explanation:

The denominator cannot be zero as this would make y 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+34=0rArrx=-34" is the asymptote"`

`"horizontal asymptotes occur as "`

`lim_(xto+-oo),ytoc"( a constant)"`

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

`y=((13x)/x)/(x/x+(34)/x)=13/(1+34/x)`

as `xto+-oo,yto13/(1+0)`

`rArry=13" 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.
graph{(13x)/(x+34) [-80, 80, -40, 40]}