Consider the function
$$f(x) = \frac{-4}{x-2}$$
To find the $x$-intercept, $y$ is set equal to $0$. However, in rational functions, the numerator is set equal to $0$. In the example above, since the numerator is constant, there is no $x$-intercept. Why is the numerator the only one taken into account when finding the $x$-intercept in rational functions?
Best Answer
The $x$-intercepts are given by solving $f(x) =0$. In your case, this is $$ - \frac{4}{x-2} =0.$$ Multiply both sides by $x-2$, you get $-4 =0$, which is nonsense. Hence, $f(x)$ has no $x$-intercepts, and thus has a horizontal asymptote at $y=0$.