Question

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

​

Answer 1

for a rational expression the vertical asymptotes occur when the denominator equals 0, in this case that will be when x + a = 0.

now, if there were to be a vertical asymptote of x = 1, that simply means that

x = 1 ==> x - 1 = 0.

meaning that a = -1.

horizontal asymptotes occur when the denominator has a higher degree than the numerator OR when both have the same degree.

when the degree of the denominator is higher, then the only horizontal asymptote occurring is y = 0.

when the degrees are the same, then the horizontal asymptote occurs at the leading terms' coefficient fraction.

now, if this expression were to have a horizontal asymptote of y = 2, that simply means

`\bf \cfrac{2x^m}{x+a}\implies \cfrac{2x^1}{1x^1+a}\implies \stackrel{\textit{horizontal asymptote}}{\cfrac{2}{1}\implies y=2}\qquad \textit{meaning m = 1}`

Answer 2

a=-1,m=1

Step-by-step explanation

The given function is

`f(x) = \frac{2 {x}^(m) }{x + a}`

For this rational function to have a horizontal asymptote at y=2, the degree of the numerator must equal the degree of the denominator.

This implies that, we must have m=1.

For the function to have a vertical asymptote at x=1, then,

`1 + a = 0`

This implies that,

`a = 0 - 1`

`a = - 1`

The correct choice is the third option.