The expression \( \frac{1}{x-3} \) represents a rational function. Here's a breakdown:

### Key Properties
1. **Domain:** The expression is undefined at \( x = 3 \), because the denominator becomes zero. Therefore, the domain is all real numbers except \( x = 3 \), which can be written as:
   \[
   x \in \mathbb{R}, \; x \neq 3
   \]

2. **Vertical Asymptote:** At \( x = 3 \), there is a vertical asymptote, as the function approaches infinity or negative infinity.

3. **Behavior as \( x \to \pm\infty \):** As \( x \) grows very large (positively or negatively), the value of \( \frac{1}{x-3} \) approaches 0, resulting in a horizontal asymptote at \( y = 0 \).

4. **Sign Analysis:**
   - For \( x > 3 \), \( x - 3 > 0 \), so \( \frac{1}{x-3} > 0 \).
   - For \( x < 3 \), \( x - 3 < 0 \), so \( \frac{1}{x-3} < 0 \).

5. **Graph:** The graph consists of two branches:
   - One in the first quadrant (for \( x > 3 \)).
   - One in the third quadrant (for \( x < 3 \)).

Would you like help sketching the graph or further analysis?  

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### Related Questions
1. What is the limit of \( \frac{1}{x-3} \) as \( x \to 3^+ \) and \( x \to 3^- \)?
2. Can you find the derivative of \( \frac{1}{x-3} \) and interpret its meaning?
3. How does \( \frac{1}{x-3} \) compare with \( \frac{1}{x+a} \) for any constant \( a \)?
4. What is the integral of \( \frac{1}{x-3} \) with respect to \( x \)?
5. How would you express \( \frac{1}{x-3} \) as a power series?

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### Tip
To analyze rational functions effectively, always check for points of discontinuity and asymptotic behavior first.

Explore the detailed analysis of the rational function 1/(x-3), including its domain, vertical asymptote, horizontal asymptote, and sign behavior. Learn how to determine the function's key properties and sketch its graph. Suitable for students in Grades 10-12.

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