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Chapter 3: Problem 37

Find the horizontal asymptote, if there is one, of the graph of each rational function. $$ f(x)=\frac{12 x}{3 x^{2}+1} $$

Short Answer

Expert verified
The horizontal asymptote of the given rational function \(f(x) = \frac{12 x}{3x^{2}+1}\) is at y=0.

Step by step solution

01

- 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.
02

- 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.
03

- 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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