Question

How do you find the vertical, horizontal and slant asymptotes of: `f(x) = (x - 3) / (x + 2)`?

Answer 1

vertical asymptote x = -2
horizontal asymptote y = 1

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 +2 = 0 `rArrx=-2" is the asymptote"`

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

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

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

`rArry=1" is the asymptote"`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1 ) Hence there are no slant asymptotes.
graph{(x-3)/(x+2) [-20, 20, -10, 10]}