Question

What are the asymptotes and removable discontinuities, if any, of `f(x)=(-x^2-2x+15)/( x-5)`?

Answer 1

`"vertical asymptote at "x=5`
`"slant asymptote "y=-x-7`

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-5=0rArrx=5" is the asymptote"`

`"Horizontal asymptotes occur if the degree of the "`
`"numerator "<=" to the degree of the denominator"`

`"this is not the case here hence there are no horizontal"`
`"asymptotes"`

`"slant asymptotes occur if the degree of the numerator"`
`>" degree of the denominator"`

`"hence there is a slant asymptote so divide"`

`-x(x-5)-7(x-5)-20`

`rArrf(x)=(-x^2-2x+15)/(x-5)=-x-7-20/(x-5)`

`"as "xto+-oo,f(x)to-x-7`

`rArry=-x-7" is the asymptote"`
graph{(-x^2-2x+15)/(x-5) [-10, 10, -5, 5]}