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.