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(i) Identify the horizontal and vertical asymptotes  t(x = 1) and (x = 5). Explain how you would find horizontal and vertical asymptotes of any rational function mathematically.

### Identification of Horizontal and Vertical Asymptotes

From the provided graph:

**Vertical Asymptotes:**
- The graph shows vertical asymptotes at \( x = 1 \) and \( x = 5 \).

**Horizontal Asymptote:**
- There is no horizontal asymptote visible in the provided portion of the graph, but the curve appears to approach \( y = 0 \) as \( x \to \pm\infty \).

### Explanation of Finding Asymptotes for Any Rational Function

A rational function is typically given by:
\[ f(x) = \frac{P(x)}{Q(x)} \]
where \( P(x) \) and \( Q(x) \) are polynomials.

**1. Vertical Asymptotes:**
- Vertical asymptotes occur where the denominator \( Q(x) \) is zero and the numerator \( P(x) \) is non-zero.
- To find vertical asymptotes, solve \( Q(x) = 0 \).
- Example: If \( f(x) = \frac{2x + 3}{(x - 1)(x - 5)} \), then set \( (x - 1)(x - 5) = 0 \), giving vertical asymptotes at \( x = 1 \) and \( x = 5 \).

**2. Horizontal Asymptotes:**
- Horizontal asymptotes describe the behavior of \( f(x) \) as \( x \to \pm\infty \).
- Compare the degrees of \( P(x) \) and \( Q(x) \):
  - If the degree of \( P(x) \) is less than the degree of \( Q(x) \), the horizontal asymptote is \( y = 0 \).
  - If the degrees of \( P(x) \) and \( Q(x) \) are equal, the horizontal asymptote is \( y = \frac{a}{b} \) where \( a \) and \( b \) are the leading coefficients of \( P(x) \) and \( Q(x) \), respectively.
  - If the degree of \( P(x) \) is greater than the degree of \( Q(x) \), there is no horizontal asymptote (there might be an oblique/slant asymptote instead).

Would you like more details or have any questions? Here are five additional questions you might be interested in:

1. How do you find oblique asymptotes of a rational function?
2. Can you explain how to sketch the graph of a rational function?
3. How do you determine the x-intercepts and y-intercepts of a rational function?
4. What is the difference between a hole and a vertical asymptote in a rational function?
5. How do you determine the behavior of a function near its asymptotes?

**Tip:** When analyzing the graph of a rational function, always start by identifying vertical and horizontal asymptotes, as they provide a framework for understanding the function's behavior.

Learn how to identify horizontal and vertical asymptotes of rational functions with detailed explanations and examples. This problem covers key concepts in calculus and algebra, including rational functions, asymptotes, and polynomial equations. Suitable for advanced high school students and college-level learners.

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