Question

For what values of `a` and `m` does `f(x) = (2x^m)/(x+a)` have a horizontal asymptote at y=2 and a vertical asymptote at x=1?

Answer 1

`a=-1" and "m=1`

Explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

`"if "a=-1`

`rArrx-1=0rArrx=1" is the asymptote"`

`"for a horizontal asymptote "y=2`

`"the degrees of the numerator and denominator must be "`
`"equal"`

`rArrm=1`

`rArrf(x)=(2x)/(x-1)`

`"horizontal asymptotes occur as"`

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

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

`f(x)=((2x)/x)/(x/x-1/x)=2/(1-1/x)`

as `xto+-oo,f(x)to2/(1-0)`

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