Find the value xequalsa where the function is discontinuous. For the point of​ discontinuity, give​ (a) f(a) if it​ exists, (b) ModifyingBelow lim With x right arrow a Superscript minus f left parenthesis x right parenthesis​, ​(c) ModifyingBelow lim With x right arrow a Superscript plus f left parenthesis x right parenthesis​, ​(d) ModifyingBelow lim With x right arrow a f left parenthesis x right parenthesis​, and​ (e) identify which conditions for continuity are not met. 
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A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical f(x)-axis labeled from negative 10 to 10 in increments of 1. A graph consists of two rays and a line segment. A ray rises from left to right passing through the point (negative 7, negative 2) to the closed point (negative 6, negative 1). A horizontal line segment extends from the closed point (negative 6, negative 1) to the open point (negative 3, negative 1). A ray rises from left to right from the open point (negative 3, negative 1) and passes through the point (negative 2, 0). There is also a closed point at (negative 3, 9).
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Part 1
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Let's break this problem down and analyze the discontinuity in the function graph.

### Step 1: Identify the point of discontinuity.
Looking at the graph description, the function seems to be discontinuous at **x = -3**. At this point, there is a jump in the function values. Specifically, the graph has:
- An **open point** at \((-3, -1)\), indicating that the function is not defined at \(x = -3\) with the value \(-1\).
- A **closed point** at \((-3, 9)\), showing that the function has the value 9 at \(x = -3\).

Therefore, \(a = -3\) is the value where the function is discontinuous.

### Step 2: Analyze the function behavior at \(x = a = -3\).

#### (a) \( f(a) \) if it exists:
The closed point at \((-3, 9)\) indicates that \(f(-3) = 9\).

#### (b) \( \lim_{{x \to a^-}} f(x) \):
The left-hand limit means the value of the function as \(x\) approaches \(-3\) from the left side. From the description, as \(x\) approaches \(-3\) from the left, the function approaches \(-1\), due to the open point at \((-3, -1)\). Hence,
\[
\lim_{{x \to -3^-}} f(x) = -1.
\]

#### (c) \( \lim_{{x \to a^+}} f(x) \):
The right-hand limit means the value of the function as \(x\) approaches \(-3\) from the right side. Based on the graph, the function also approaches \(-1\) from the right side. Hence,
\[
\lim_{{x \to -3^+}} f(x) = -1.
\]

#### (d) \( \lim_{{x \to a}} f(x) \):
The overall limit as \(x\) approaches \(-3\) only exists if the left-hand and right-hand limits are equal. Since both limits are equal to \(-1\), we have:
\[
\lim_{{x \to -3}} f(x) = -1.
\]

#### (e) Which conditions for continuity are not met?
For a function to be continuous at a point \(x = a\), three conditions must be satisfied:
1. \( f(a) \) must exist.
2. \( \lim_{{x \to a}} f(x) \) must exist.
3. \( f(a) = \lim_{{x \to a}} f(x) \).

In this case:
- \( f(-3) = 9 \),
- \( \lim_{{x \to -3}} f(x) = -1 \).

Since \( f(-3) \neq \lim_{{x \to -3}} f(x) \), the function is **discontinuous at \(x = -3\)**. The condition that is not met is that \(f(a)\) does not equal the limit of the function as \(x\) approaches \(a\).

### Summary:
- \(a = -3\),
- \(f(-3) = 9\),
- \(\lim_{{x \to -3^-}} f(x) = -1\),
- \(\lim_{{x \to -3^+}} f(x) = -1\),
- \(\lim_{{x \to -3}} f(x) = -1\),
- Discontinuity occurs because \(f(-3) \neq \lim_{{x \to -3}} f(x)\).

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Would you like to dive deeper into any of these details?

Here are 5 related questions to consider:
1. What are the different types of discontinuities a function can have?
2. How do you determine if a function is continuous at a point?
3. What is the significance of left-hand and right-hand limits?
4. How can a piecewise function be written for this problem?
5. What are the conditions for a function to have a removable discontinuity?

**Tip**: When analyzing discontinuities, always check if the left-hand and right-hand limits are equal to identify the type of discontinuity.

Find the value x=a where the function is discontinuous. For the point of discontinuity, give (a) f(a) if it exists, (b) lim x->a- f(x), (c) lim x->a+ f(x), (d) lim x->a f(x), and (e) identify which conditions for continuity are not met. A graph consists of two rays and a line segment, indicating a discontinuity at x = -3.

Discover how to find and analyze discontinuities in a function. This problem breaks down limits, left-hand and right-hand limits, and checks conditions for continuity at x = -3. Learn the behavior of piecewise functions with step-by-step explanations. Ideal for students in Grades 11-12 studying pre-calculus or calculus.

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