Question

given the function below:
f(x) = \frac{x}{x - 3}
the horizontal asymptote for this function is ______.
\bigcirc there is no horizontal asymptote.
\bigcirc y=1
\bigcirc y=3
\bigcirc y=0

Explanation:

Step1: Recall horizontal asymptote rule

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), where \( N(x) \) and \( D(x) \) are polynomials. If the degrees of \( N(x) \) and \( D(x) \) are equal, the horizontal asymptote is the ratio of the leading coefficients.
Here, \( N(x)=x \) (degree 1, leading coefficient 1) and \( D(x)=x - 3 \) (degree 1, leading coefficient 1).

Step2: Apply the rule

Since the degrees are equal, the horizontal asymptote is \( y=\frac{\text{leading coefficient of }N(x)}{\text{leading coefficient of }D(x)}=\frac{1}{1} = 1 \).

Answer:

\( y = 1 \)