Question

find equations for the horizontal asymptotes, if any, for the following rational function.
$f(x) = \frac{3}{x + 2}$
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Explanation:

Step1: Recall Horizontal Asymptote Rules

For a rational function \( f(x) = \frac{N(x)}{D(x)} \), where \( \text{deg}(N(x)) \) is the degree of the numerator and \( \text{deg}(D(x)) \) is the degree of the denominator:

• If \( \text{deg}(N(x)) < \text{deg}(D(x)) \), the horizontal asymptote is \( y = 0 \).
• If \( \text{deg}(N(x)) = \text{deg}(D(x)) \), the horizontal asymptote is \( y = \frac{\text{leading coefficient of } N(x)}{\text{leading coefficient of } D(x)} \).
• If \( \text{deg}(N(x)) > \text{deg}(D(x)) \), there is no horizontal asymptote (but there may be an oblique asymptote).

Step2: Determine Degrees of Numerator and Denominator

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

• The numerator \( N(x) = 3 \) is a constant polynomial, so its degree \( \text{deg}(N(x)) = 0 \).
• The denominator \( D(x) = x + 2 \) is a linear polynomial, so its degree \( \text{deg}(D(x)) = 1 \).

Step3: Apply the Rule

Since \( \text{deg}(N(x)) = 0 < \text{deg}(D(x)) = 1 \), by the rule for horizontal asymptotes, the horizontal asymptote is \( y = 0 \).

Answer:

\( y = 0 \)