To find the horizontal asymptote of the function

$f(x) = \frac{2x^2 + 6x}{x^2 - x}$

we need to examine the degrees of the polynomials in the numerator and denominator.

Step 1: Degree of the numerator and denominator

• The degree of the numerator $2x^2 + 6x$ is 2 (since the highest power of $x$ is $x^2$).
• The degree of the denominator $x^2 - x$ is also 2 (since the highest power of $x$ is $x^2$).

Step 2: Horizontal asymptote rule

When the degrees of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients.

• The leading coefficient of the numerator is $2$ (from $2x^2$).
• The leading coefficient of the denominator is $1$ (from $x^2$).

Step 3: Conclusion

Thus, the horizontal asymptote is:

$y = \frac{2}{1} = 2$

So, the horizontal asymptote of the given function is $y = 2$.

Let me know if you need more details on how to find other types of asymptotes or have additional questions!

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Here are some follow-up questions you might find interesting:

1. How do you find the vertical asymptote of this function?
2. What happens if the degrees of the numerator and denominator are different?
3. Can a function have both horizontal and vertical asymptotes at the same time?
4. What is the significance of horizontal asymptotes in real-world scenarios?
5. How do you handle horizontal asymptotes when the degree of the numerator is greater than the denominator?

Tip: For rational functions, always start by comparing the degrees of the numerator and denominator to determine the horizontal asymptote!