Question

How do you identify all asymptotes or holes and intercepts for `f(x)=x/(3x^2+5x)`?

Answer 1

`"see explanation"`

Explanation:

`"factorise the denominator"`

`f(x)=cancel(x)/(cancel(x)(3x+5))=1/(3x+5)`

`"since the factor x has been cancelled from the numerator/"`
`"denominator this indicates a hole at"(0,1/5)`

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 "3x+5=0rArrx=-5/3" is the asymptote"`

`"horizontal asymptotes occur as"`

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

`"divide numerator/denominator by x"`

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

`"as "xto+-oo,f(x)to0/(3+0)`

`rArry=0" is the asymptote"`
graph{1/(3x+5) [-10, 10, -5, 5]}