To find horizontal asymptotes you have to find the value that the function has a limit when x is inf
[tex]\lim_{x\to\infty}\frac{\left(x+6\right)}{x(x+8)}[/tex]
Divide for the denominator whit the greatest potency to solve:
[tex]\lim _{x\to \infty \:}\left(\frac{\frac{x}{x+8}+\frac{6}{x+8}}{x}\right)[/tex]
Use limits properties:
[tex]\lim_{x\to a}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{\lim_{x\to a}f\left(x\right)}{\lim_{x\to a}g\left(x\right)}[/tex][tex]=\frac{\lim _{x\to \infty \:}\left(\frac{x}{x+8}+\frac{6}{x+8}\right)}{\lim _{x\to \infty \:}\left(x\right)}[/tex]
Separate terms:
[tex]\lim_{x\to\infty\:}\left(\frac{x}{x+8}\right)=\lim_{x\to\infty\:}\left(\frac{1}{1+\frac{8}{x}}\right)=\frac{\lim_{x\to\infty\:}\left(1\right)}{\lim_{x\to\infty\:}\left(1+\frac{8}{x}\right)}=\frac{1}{1}=1[/tex]
[tex]\lim_{x\to\infty\:}\left(\frac{6}{x+8}\right)=6\cdot\frac{\lim_{x\to\infty\:}\left(1\right)}{\lim_{x\to\infty\:}\left(x+8\right)}=6\cdot\frac{1}{\infty\:}=\frac{6}{\infty}=0[/tex]
[tex]1+0=1[/tex]
[tex]\lim_{x\to\infty\:}\left(x\right)=\infty[/tex][tex]\lim_{x\to\infty\:}\left(\frac{x+6}{x\left(x+8\right)}\right)=\frac{1}{\infty}=0[/tex]
So the function has a horizontal asymptote in
y=0
