Problem 37
Question
Find the horizontal asymptote, if there is one, of the graph of each rational function. $$ f(x)=\frac{12 x}{3 x^{2}+1} $$
Step-by-Step Solution
Verified Answer
The horizontal asymptote of the given rational function \(f(x) = \frac{12 x}{3x^{2}+1}\) is at y=0.
1Step 1 - Understand the Polynomials
Observe the function \( f(x) = \frac{12 x}{3x^{2}+1} \) which is in the form of \( \frac{P(x)}{Q(x)} \). Here, P(x) = 12x is a polynomial of degree 1 and Q(x) = \(3x^{2}+1\) is a polynomial of degree 2.
2Step 2 - Applying the Asymptote Rule
Since the degree of the denominator Q(x) is greater than the degree of the numerator P(x), according to the rule, there will be a horizontal asymptote at y=0.
3Step 3 - Conclusion
With the information from step 2, we conclude that for the function \(f(x) = \frac{12 x}{3x^{2}+1}\), the horizontal asymptote is at y=0.
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