To find the horizontal and vertical asymptotes of the function $f(x) = \frac{2x^2 - 5x + 3}{x^2 + 5x}$, we follow these steps:

Horizontal Asymptotes

Horizontal asymptotes are found by considering the limits of $f(x)$ as $x$ approaches $\infty$ or $-\infty$.

1. Degree of the numerator (N) and denominator (D):

   • Degree of $N(x) = 2x^2 - 5x + 3$ is 2.
   • Degree of $D(x) = x^2 + 5x$ is 2.

2. Comparing degrees:

   • If the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients.

Here, both degrees are 2, and the leading coefficients are:

• Numerator: 2
• Denominator: 1

Therefore, the horizontal asymptote is: $y = \frac{2}{1} = 2$

Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero.

1. Set the denominator to zero: $x^2 + 5x = 0$ $x(x + 5) = 0$

2. Solve for $x$: $x = 0$ $x = -5$

Conclusion

The vertical asymptotes are at $x = 0$ and $x = -5$. The horizontal asymptote is at $y = 2$.

Would you like more details or have any questions?

Here are some more questions you might consider:

1. How to determine slant asymptotes for rational functions?
2. What are the steps to find oblique asymptotes if they exist?
3. Can you explain how to find holes in the graph of a rational function?
4. How do we determine the behavior of a function near its asymptotes?
5. What are the steps to sketch the graph of a rational function?

Tip: When analyzing asymptotes, always compare the degrees of the numerator and the denominator to quickly identify potential horizontal asymptotes.