https://file.aiquickdraw.com/mathbot/file/1726091834133mrkr4n0a.jpg

Let's analyze the graph and answer the questions provided.

1. \( f(-1) \)

To find \( f(-1) \), we look at the y-coordinate of the point where \( x = -1 \). From the graph, at \( x = -1 \), \( y = 3 \). Therefore, \( f(-1) = 3 \).

2. \( \lim_{x \to -1^-} f(x) \)

The limit of \( f(x) \) as \( x \) approaches \(-1\) from the left. From the graph, as \( x \) approaches \(-1\) from the left, \( f(x) \) approaches \( 3 \). Therefore, \( \lim_{x \to -1^-} f(x) = 3 \).

3. \( \lim_{x \to -1^+} f(x) \)

The limit of \( f(x) \) as \( x \) approaches \(-1\) from the right. From the graph, as \( x \) approaches \(-1\) from the right, \( f(x) \) also approaches \( 3 \). Therefore, \( \lim_{x \to -1^+} f(x) = 3 \).

4. \( \lim_{x \to -1} f(x) \)

Since both the left-hand limit and the right-hand limit as \( x \) approaches \(-1\) are equal and both are \( 3 \), the limit of \( f(x) \) as \( x \) approaches \(-1\) exists and is \( 3 \). Therefore, \( \lim_{x \to -1} f(x) = 3 \).

To summarize, the answers are:

1. \( f(-1) = 3 \)
2. \( \lim_{x \to -1^-} f(x) = 3 \)
3. \( \lim_{x \to -1^+} f(x) = 3 \)
4. \( \lim_{x \to -1} f(x) = 3 \)

Would you like any further details or have any questions about these solutions? Here are some related questions you might find useful:

1. What are the definitions of left-hand limit and right-hand limit?
2. How do we determine if a limit exists at a point?
3. What is the difference between a limit and a function value at a point?
4. Can a function be discontinuous at a point where its limit exists?
5. How do you identify and interpret discontinuities in a graph?

**Tip:** Always compare the left-hand and right-hand limits at a point to determine if the overall limit exists.

Consider the graph of y = f(x) shown. Find f(-1), lim(x→−1−)f(x), lim(x→−1+)f(x), and lim(x→−1)f(x) based on the graph.

This example uses the graph of a function to determine key concepts like the function value f(-1) and the limits as x approaches -1 from both sides. A useful resource for students learning about limits, continuity, and graphs in Grades 10-12.

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