Question

give an example of a rational function that has a horizontal asymptote of y=1/2 and vertical asymptote at x=-3

Answer 1

The horizontal asymptote of a function is the value of y when x -> ∞

$\lim _{x\to\infty}f(x)=y$

and the vertical asymptote of a function is the x-value when y -> ∞.

For a rational function, the y approaches infinity when the denominator vanishes. In other words, when in q/p, p --> 0.

Using this fact we see that the denominator of our function must vanish at x = -3. This happens when the denominator has the form

$\frac{1}{x+3}$

because

$\lim _{x\to-3}\frac{1}{x+3}=\infty$

is the vertical asymptote.

For the horizontal asymptote, the function can be of the form

$f(x)=\frac{x}{2(x+3)}$

because

$\textcolor{#FF7968}{\lim _{x\to\infty}\frac{x}{2(x+3)}=\frac{1}{2}}$

Hence the function is

$f(x)=\frac{x}{2(x+3)}$