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Let's go through the questions one by one and work on solutions or guidance. 

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### Problem 1:
You are given the function \( f(x) \):

\[
f(x) = \begin{cases} 
\frac{(x-2.5)\sin\left(\frac{1}{x-2.5}\right)}{x-2.5} & \text{for } x \neq 2.5 \\
b & \text{for } x = 2.5
\end{cases}
\]

#### (A) What sort of discontinuity does \( f \) have at \( x = 2.5 \)? Justify your answer using limits.

- The function has a piecewise form, with different expressions depending on whether \( x = 2.5 \) or not.
- To determine the type of discontinuity, compute \( \lim_{x \to 2.5} f(x) \). If the limit exists and equals \( f(2.5) \), the function is continuous. Otherwise, there's a discontinuity.

#### (B) For what value of \( b \), if any, is \( f \) continuous at \( x = 2.5 \)?

- Continuity at \( x = 2.5 \) requires that \( \lim_{x \to 2.5} f(x) = f(2.5) = b \).
- Find \( \lim_{x \to 2.5} f(x) \) using the limit of the non-piecewise part and set it equal to \( b \) to solve.

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### Problem 2:
You are given \( g(x) = \frac{\sqrt{x}}{x-87} \).

#### (A) What sort of discontinuities does \( g \) have at \( x = 0 \) and \( x = 87 \)? Explain your answer using limits.

- Identify where the denominator equals zero, as this indicates possible discontinuities. Calculate one-sided limits at \( x = 0 \) and \( x = 87 \) to determine the type of discontinuity.

#### (B) What is the domain of \( g(x) \)?

- The function involves a square root and a denominator, which affect the domain. Set conditions for \( x \) to ensure \( g(x) \) is real and defined.

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### Problem 3:
Given \( g(x) = \begin{cases} 
\frac{\sin(12x)}{3x} & \text{for } x \neq 0 \\
c & \text{for } x = 0
\end{cases} \).

#### (A) Find \( \lim_{x \to 0} g(x) \).

- Use L'Hopital's Rule or trigonometric limits to evaluate \( \lim_{x \to 0} g(x) \).

#### (B) For what value of \( c \) is \( g \) continuous at \( x = 0 \)?

- Set \( c = \lim_{x \to 0} g(x) \) for continuity at \( x = 0 \).

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### Problem 4:
You are given the function:

\[
f(x) = \begin{cases} 
x^2 - 4x + \frac{17}{3} & \text{if } x \leq 3 \\
x^2 + 3x & \text{if } x > 3
\end{cases}
\]

#### (A) Show that \( f \) has a zero between \( x = -3 \) and \( x = -2 \) using the Intermediate Value Theorem (IVT).

- Check \( f(-3) \) and \( f(-2) \). If one is positive and the other negative, IVT guarantees a zero between them.

#### (B) Is \( f \) continuous at \( x = 3 \)? Give a reason for your answer.

- Compute \( \lim_{x \to 3^-} f(x) \), \( \lim_{x \to 3^+} f(x) \), and compare them with \( f(3) \).

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### Problem 5:
This asks to draw a function that satisfies certain conditions related to the Intermediate Value Theorem (IVT). You need to sketch graphs based on the given conditions.

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### Problem 6:
The function \( f(x) = -5.5 + |x - 4| \) is given, along with a table of values.

#### (A) What is the fewest possible number of times \( f'(x) = 0 \)?

- Analyze the behavior of \( f(x) \) based on the table and function properties to determine critical points.

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Do you want to solve specific questions or go over detailed solutions for any part? Here's a set of expanded questions based on the current information:

1. How do you determine the type of discontinuity from limits?
2. How do we apply L'Hopital's Rule to solve \( \lim_{x \to 0} \frac{\sin(12x)}{3x} \)?
3. What are the steps for proving the existence of a zero using the Intermediate Value Theorem?
4. How can you confirm if a piecewise function is continuous at a point?
5. How does the derivative affect the behavior of a function in a table?

**Tip:** Always check both left-hand and right-hand limits when determining the continuity of a piecewise function.

Please record your answers on a separate sheet of paper. Several calculus problems involving continuity, limits, and intermediate value theorem are presented.

Explore solutions to AP Calculus AB problems on continuity, limits, and the Intermediate Value Theorem. Learn how to determine discontinuities, find function domains, and apply calculus rules with step-by-step explanations.

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