Question

find all horizontal asymptotes of the following function.
$f(x) = \frac{3(x + 3)}{2(x + 10)(x + 3)}$
answer attempt 1 out of 2
one horizontal asymptote

Explanation:

Step1: Simplify the function

First, we can cancel out the common factor \((x + 3)\) in the numerator and the denominator, provided that \(x
eq - 3\). So the function simplifies to \(f(x)=\frac{3}{2(x + 10)}\) for \(x
eq - 3\).

Step2: Analyze the degrees of numerator and denominator

For a rational function \(y = \frac{N(x)}{D(x)}\), where \(N(x)\) is the numerator and \(D(x)\) is the denominator:

• If the degree of \(N(x)\) is less than the degree of \(D(x)\), the horizontal asymptote is \(y = 0\).
• If the degree of \(N(x)\) is equal to the degree of \(D(x)\), the horizontal asymptote is \(y=\frac{\text{leading coefficient of }N(x)}{\text{leading coefficient of }D(x)}\).
• If the degree of \(N(x)\) is greater than the degree of \(D(x)\), there is no horizontal asymptote (but there may be an oblique asymptote).

In our simplified function \(f(x)=\frac{3}{2(x + 10)}=\frac{3}{2x+20}\), the degree of the numerator \(N(x) = 3\) (a constant polynomial, degree 0) and the degree of the denominator \(D(x)=2x + 20\) (degree 1). Since \(0<1\) (degree of numerator < degree of denominator), the horizontal asymptote is \(y = 0\).

Answer:

\(y = 0\)