Question

What are the asymptotes and removable discontinuities, if any, of `f(x)=(x-3)/(2x+5)`?

Answer 1

`"vertical asymptote at "x=-5/2`
`"horizontal asymptote at "y=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 "2x+5=0rArrx=-5/2" is the asymptote"`

`"Horizontal asymptotes occur as"`

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

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

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

`"as "xto+-oo,f(x)to(1-0)/(2+0)`

`rArry=1/2" is the asymptote"`

`"Removable discontinuities occur when common"`
`"factors are cancelled from the numerator/denominator."`
`"this is not the case here hence there are no removable"`
`"discontinuities"`
graph{(x-3)/(2x+5) [-10, 10, -5, 5]}