https://file.aiquickdraw.com/mathbot/file/1726457118557jiwcntkc.jpg

The image shows a graph of a piecewise function \( f(x) \), and the question asks for the limit of \( f(x) \) as \( x \to -1 \), i.e., \( \lim_{x \to -1} f(x) \). Let's break this down.

1. The question asks to **determine** the value of the limit \( \lim_{x \to -1} f(x) \) based on the provided graph.
2. We can solve this by observing the behavior of the graph near \( x = -1 \).

### Step-by-Step Analysis:
- **Left-hand limit**: As \( x \) approaches -1 from the left, the curve approaches \( y = -3 \).
- **Right-hand limit**: As \( x \) approaches -1 from the right, the curve approaches \( y = -3 \).
  
Since both the left-hand and right-hand limits are equal and they approach \( -3 \), we conclude that:
\[
\lim_{x \to -1} f(x) = -3
\]

### Answer:
The correct answer is **\( -3 \)**, which corresponds to option (a) in the provided multiple-choice options.

Would you like further details or explanations?

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Here are 5 related questions for further practice:
1. What is the limit \( \lim_{x \to 1} f(x) \) from the same graph?
2. How do you calculate the right-hand and left-hand limits separately?
3. What does it mean if the left-hand and right-hand limits are not equal?
4. How do we determine if a function is continuous at a point?
5. What is the formal definition of a limit in calculus?

**Tip:** When solving limit problems graphically, always check both sides of the point where the limit is being evaluated to ensure that the left-hand and right-hand limits are equal.

Determine the limit of the function f(x) as x approaches -1, based on the graph of the piecewise function f(x).

Find the limit of the function f(x) as x approaches -1 using a graph. The problem involves analyzing left-hand and right-hand limits to determine the correct answer. Suitable for high school students in Grades 10-12.

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