Question

What are the asymptotes and removable discontinuities, if any, of `f(x)= (x^2 - 2x - 3) /(-4x )`?

Answer 1

`"vertical asymptote at " x=0`
`"oblique asymptote "y=-1/4x+1/2`

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

Oblique / slant asymptotes occur when the degree of the numerator is > degree of the denominator. This is the case here ( numerator-degree 2, denominator- degree 1)

`"dividing gives"`

`f(x)=x^2/(-4x)-(2x)/(-4x)-3/(-4x)=-1/4x+1/2+3/(4x)`

`"as "xto+-oo,f(x)to-1/4x+1/2`

`rArry=-1/4x+1/2" is the asymptote"`
graph{(x^2-2x-3)/(-4x) [-10, 10, -5, 5]}