Horizontal asymptotes, if any, can be found by exploring limit of $f\left(x\right)$ as $x→∞$ and limit of $f\left(x\right)$ as $x→−∞$.
If either, or both, are constants, then the constant(s) is/are the horizontal asymptote(s).

Vertical asymptotes and holes occur where the denominator equals $0$. If the zero on the denominator 'cancels out', then that is the $x$-value of the hole. Find the $y$-value too. If the zero in the denominator remains, then that is the vertical asymptote.

Now that you have found all holes and asymptotes... Use the $x$-values of all holes and vertical asymptotes to restrict the domain. Also consider endpoint(s). Use the $y$-values of all holes and horizontal asymptotes to restrict the range. Also, consider endpoint(s).