Question

How do you find the vertical, horizontal or slant asymptotes for `(2x)/(x^2+16)`?

Answer 1

`"horizontal asymptote at " y=0`

Explanation:

`f(x)=(2x)/(x^2+16)`

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^2+16=0rArrx^2=-16`

`"this has no real solutions hence there are no"`
`"vertical asymptotes"`

`"horizontal asymptotes occur as"`

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

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

`f(x)=((2x)/x^2)/(x^2/x^2+16/x^2)=(2/x)/(1+16/x^2)`

as `xto+-oo,f(x)to0/(1+0)`

`rArry=0" is the asymptote"`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 1, denominator-degree 2) hence there are no slant asymptotes.
graph{(2x)/(x^2+16) [-10, 10, -5, 5]}