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Let's go through the problem step by step.

The function \( f(x) \) is defined piecewise as:

\[
f(x) =
\begin{cases}
-x + b, & \text{if } x < 3 \\
1, & \text{if } x = 3 \\
-\frac{1}{x - b} + 0, & \text{if } x > 3 \text{ (and } x \neq b)
\end{cases}
\]

### Part (a): Continuity at \( x = 3 \)

For \( f(x) \) to be continuous at \( x = 3 \), the left-hand limit, the right-hand limit, and the value of the function at \( x = 3 \) must all be equal.

1. **Left-hand limit as \( x \to 3^- \)**:
   \[
   \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (-x + b) = -3 + b
   \]

2. **Right-hand limit as \( x \to 3^+ \)**:
   \[
   \lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} \left( -\frac{1}{x - b} \right)
   \]

   However, since we are evaluating the limit as \( x \) approaches 3, we must carefully consider the value of \( b \). If \( b = 3 \), the denominator would go to zero, indicating a possible discontinuity.

3. **Value at \( x = 3 \)**:
   \[
   f(3) = 1
   \]

For continuity at \( x = 3 \):
\[
\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3) = 1
\]

Thus:
\[
-3 + b = 1 \quad \Rightarrow \quad b = 4
\]

### Part (b): Removable Discontinuity at \( x = 3 \)

For \( f(x) \) to have a removable discontinuity at \( x = 3 \), the left-hand and right-hand limits must be equal, but they do not necessarily need to equal the value of the function at \( x = 3 \).

From part (a), we already know:
\[
\lim_{x \to 3^-} f(x) = -3 + b
\]
\[
\lim_{x \to 3^+} f(x) = -\frac{1}{x - b} \quad (\text{for values of } b \neq 3)
\]

In this case, we need the left-hand and right-hand limits to match. So:
\[
-3 + b = 1
\]

Therefore, the value of \( b \) remains \( b = 4 \) to match both limits, and hence, it removes the discontinuity at \( x = 3 \).

Would you like further details on these steps or have any other questions? 

Here are 5 related questions to think about:
1. What if \( b = 3 \)? How does the function behave then?
2. What is the significance of a removable discontinuity?
3. Can we calculate the derivative of this function at \( x = 3 \)?
4. How do we classify discontinuities besides removable ones?
5. How would the function change if the condition for \( x = 3 \) was removed?

**Tip**: Always check both the left and right-hand limits for continuity and discontinuity problems; sometimes, one side may exhibit a different type of behavior.

For what value(s) of b is the function f(x) continuous at x = 3 and does f(x) have a removable discontinuity at x = 3?

Explore how to find the value of b that makes a piecewise function continuous and identify when it has a removable discontinuity at x = 3. This problem involves limits and the concept of continuity in higher-level calculus.

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