Question

sketch a possible graph of a function that satisfies the conditions below. determine whether f is continuous at x = 3. ( f(3) = 2; limlimits_{x \to 3^-} f(x) = -2; limlimits_{x \to 3^+} f(x) = 2 ) choose the correct graph below. ( \bigcirc ) a. graph ( \bigcirc ) b. graph ( \bigcirc ) c. graph ( \bigcirc ) d. graph

Explanation:

To determine the correct graph and continuity at \( x = 3 \), we analyze the given conditions:

Step 1: Analyze the Left-Hand Limit (\( \lim_{x \to 3^-} f(x) \))

The left-hand limit as \( x \) approaches 3 is \( -2 \). This means the graph should approach \( y = -2 \) from the left of \( x = 3 \).

Step 2: Analyze the Right-Hand Limit (\( \lim_{x \to 3^+} f(x) \))

The right-hand limit as \( x \) approaches 3 is \( 2 \). This means the graph should approach \( y = 2 \) from the right of \( x = 3 \).

Step 3: Analyze the Function Value (\( f(3) \))

The function value at \( x = 3 \) is \( f(3) = 2 \). This means there is a point at \( (3, 2) \).

Step 4: Determine Continuity at \( x = 3 \)

For a function to be continuous at a point \( x = a \), three conditions must be met:

1. \( f(a) \) is defined.
2. \( \lim_{x \to a} f(x) \) exists (i.e., \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) \)).
3. \( \lim_{x \to a} f(x) = f(a) \).

Here:

• \( f(3) = 2 \) (defined).
• \( \lim_{x \to 3^-} f(x) = -2 \) and \( \lim_{x \to 3^+} f(x) = 2 \). Since \( -2

eq 2 \), the limit \( \lim_{x \to 3} f(x) \) does not exist. Thus, the function is not continuous at \( x = 3 \).

Step 5: Identify the Correct Graph

We need a graph where:

• As \( x \to 3^- \), the graph approaches \( y = -2 \).
• As \( x \to 3^+ \), the graph approaches \( y = 2 \).
• There is a point at \( (3, 2) \).

Looking at the options, the correct graph should have:

• A left-hand approach to \( y = -2 \) near \( x = 3 \).
• A right-hand approach to \( y = 2 \) near \( x = 3 \).
• A solid dot at \( (3, 2) \) (since \( f(3) = 2 \)).

From the options, the graph that matches these conditions is Option C (assuming the visual cues: left side approaches \( -2 \), right side approaches \( 2 \), and a solid dot at \( (3, 2) \)).

Continuity at \( x = 3 \)

Since \( \lim_{x \to 3^-} f(x)
eq \lim_{x \to 3^+} f(x) \), the limit \( \lim_{x \to 3} f(x) \) does not exist. Therefore, \( f \) is not continuous at \( x = 3 \).

Final Answer

The correct graph is C. The function \( f \) is not continuous at \( x = 3 \) because the left-hand limit (\( -2 \)) does not equal the right-hand limit (\( 2 \)), so the overall limit as \( x \to 3 \) does not exist.

Answer:

To determine the correct graph and continuity at \( x = 3 \), we analyze the given conditions:

Step 1: Analyze the Left-Hand Limit (\( \lim_{x \to 3^-} f(x) \))

The left-hand limit as \( x \) approaches 3 is \( -2 \). This means the graph should approach \( y = -2 \) from the left of \( x = 3 \).

Step 2: Analyze the Right-Hand Limit (\( \lim_{x \to 3^+} f(x) \))

The right-hand limit as \( x \) approaches 3 is \( 2 \). This means the graph should approach \( y = 2 \) from the right of \( x = 3 \).

Step 3: Analyze the Function Value (\( f(3) \))

The function value at \( x = 3 \) is \( f(3) = 2 \). This means there is a point at \( (3, 2) \).

Step 4: Determine Continuity at \( x = 3 \)

For a function to be continuous at a point \( x = a \), three conditions must be met:

1. \( f(a) \) is defined.
2. \( \lim_{x \to a} f(x) \) exists (i.e., \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) \)).
3. \( \lim_{x \to a} f(x) = f(a) \).

Here:

• \( f(3) = 2 \) (defined).
• \( \lim_{x \to 3^-} f(x) = -2 \) and \( \lim_{x \to 3^+} f(x) = 2 \). Since \( -2

eq 2 \), the limit \( \lim_{x \to 3} f(x) \) does not exist. Thus, the function is not continuous at \( x = 3 \).

Step 5: Identify the Correct Graph

We need a graph where:

• As \( x \to 3^- \), the graph approaches \( y = -2 \).
• As \( x \to 3^+ \), the graph approaches \( y = 2 \).
• There is a point at \( (3, 2) \).

Looking at the options, the correct graph should have:

• A left-hand approach to \( y = -2 \) near \( x = 3 \).
• A right-hand approach to \( y = 2 \) near \( x = 3 \).
• A solid dot at \( (3, 2) \) (since \( f(3) = 2 \)).

From the options, the graph that matches these conditions is Option C (assuming the visual cues: left side approaches \( -2 \), right side approaches \( 2 \), and a solid dot at \( (3, 2) \)).

Continuity at \( x = 3 \)

Since \( \lim_{x \to 3^-} f(x)
eq \lim_{x \to 3^+} f(x) \), the limit \( \lim_{x \to 3} f(x) \) does not exist. Therefore, \( f \) is not continuous at \( x = 3 \).

Final Answer

The correct graph is C. The function \( f \) is not continuous at \( x = 3 \) because the left-hand limit (\( -2 \)) does not equal the right-hand limit (\( 2 \)), so the overall limit as \( x \to 3 \) does not exist.