To find the horizontal asymptotes of the given function:

$f(x) = \frac{3x - 2}{(x + 5)(3x - 2)},$

we analyze the behavior of the function as $x \to \infty$ and $x \to -\infty$.

Step 1: Identify the degree of numerator and denominator

The degree of the numerator $3x - 2$ is 1 (the highest power of $x$ is $x^1$).

The degree of the denominator is 2, since multiplying $(x+5)$ and $(3x-2)$ results in $3x^2 + \dots$, with the highest power being $x^2$.

Step 2: Compare degrees

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
• If the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there are no horizontal asymptotes (instead, there may be an oblique/slant asymptote).

Step 3: Conclusion

In this case, since the numerator has degree 1 and the denominator has degree 2, the numerator’s degree is less than the denominator’s degree.

Therefore, the horizontal asymptote is: $y = 0$

Would you like a detailed analysis or have further questions?

5 Related Questions:

1. What is the significance of comparing degrees in horizontal asymptotes?
2. How would the asymptotes change if the numerator and denominator had equal degrees?
3. What is the difference between horizontal and vertical asymptotes in a function?
4. How would you find slant (oblique) asymptotes for rational functions?
5. Can rational functions have more than one horizontal asymptote?

Tip:

Always focus on the highest powers of $x$ in the numerator and denominator to quickly determine the horizontal asymptote.