Question

What are the asymptotes and removable discontinuities, if any, of `f(x)= (x+3) / (x(x-5))`?

Answer 1

`"vertical asymptote at "x=0" and "x=5`
`"horizontal asymptote at "y=0`

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 values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

`"solve "x(x-5)=0rArrx=0,x=5" are the asymptotes"`

`"horizontal asymptotes occur as"`

`lim_(xto+-0),f(x)toc" ( a constant)"`

`"divide terms on numerator/denominator by the highest"`
`"power of x that is "x^2`

`f(x)=(x/x^2+3/x^2)/(x^2/x^2-5/x^2)=(1/x+3/x^2)/(1-5/x^2)`

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

`y=0" is the asymptote"`

`"removable discontinuities occur when a common factor is"`
`"cancelled from the numerator/denominator. This is not"`
`"the case here hence there are no removable discontinuities"`
graph{(x+3)/(x(x-5)) [-10, 10, -5, 5]}