Sketch the graph of a function that has a hole/removable discontinuity at x = 1, a
jump discontinuity at x = 2 and a vertical asymptote/infinite discontinuity at x = 3.
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To sketch a function with the given discontinuities, follow these conditions:  

1. **Hole (removable discontinuity) at \( x = 1 \):**  
   - The function is defined everywhere except at \( x = 1 \).  
   - There is a point missing (a hole) at \( x = 1 \).  
   - The function may have a limit at \( x = 1 \), but it does not take that value.  

2. **Jump discontinuity at \( x = 2 \):**  
   - The left-hand limit (\(\lim_{x \to 2^-} f(x)\)) and right-hand limit (\(\lim_{x \to 2^+} f(x)\)) exist but are not equal.  
   - This creates a sudden jump in the graph at \( x = 2 \).  

3. **Vertical asymptote at \( x = 3 \) (infinite discontinuity):**  
   - The function approaches infinity (\( +\infty \) or \( -\infty \)) as \( x \) approaches 3.  
   - The graph gets arbitrarily large or small near \( x = 3 \).  

### Possible Function Definition:
A function that satisfies these conditions could be:

\[
f(x) =
\begin{cases} 
\frac{x^2 - 1}{x - 1}, & x \neq 1, x < 2 \quad \text{(hole at \( x = 1 \))} \\
2, & x = 1 \quad \text{(hole, value not included in function)} \\
3, & x = 2^- \quad \text{(left-hand limit at \( x = 2 \))} \\
1, & x = 2^+ \quad \text{(right-hand limit at \( x = 2 \))} \\
\frac{1}{x - 3}, & x > 2, x \neq 3 \quad \text{(vertical asymptote at \( x = 3 \))}
\end{cases}
\]

### Sketching the Graph:
- **At \( x = 1 \),** draw a small open circle where the hole is.  
- **At \( x = 2 \),** show a jump by plotting different points for the left and right limits.  
- **At \( x = 3 \),** show the function approaching positive or negative infinity on either side of the vertical asymptote.  

Would you like a visual sketch of this function?  

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### Related Questions:
1. What is the formal definition of removable and jump discontinuities?
2. How do you find vertical asymptotes of a rational function?
3. What are the differences between discontinuities in piecewise functions and rational functions?
4. Can a function have more than one removable discontinuity?
5. How do limits help classify discontinuities?

**Tip:** A removable discontinuity occurs when a function is undefined at a point, but the limit exists. If you redefine the function at that point, continuity can be restored.

Sketch the graph of a function that has a hole/removable discontinuity at x = 1, a jump discontinuity at x = 2 and a vertical asymptote/infinite discontinuity at x = 3.

Learn how to sketch the graph of a function with a hole (removable discontinuity) at x = 1, a jump discontinuity at x = 2, and a vertical asymptote at x = 3. This problem covers key concepts in calculus such as discontinuities and piecewise functions. Suitable for high school students (Grades 11-12).

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