https://file.aiquickdraw.com/mathbot/file/1725914663893mqehp04q.jpg

Let's analyze the graph and solve the limits one by one based on the visual information provided. The graph shows the function \( y = f(x) \) over the interval \( x = 0 \) to \( x = 5 \).

### Part a) \( \lim_{x \to 1^-} f(x) \)
This is the limit as \( x \) approaches 1 from the left. Looking at the graph:
- As \( x \) approaches 1 from the left, \( f(x) \) gets closer to 2.
Thus, \( \lim_{x \to 1^-} f(x) = 2 \).

### Part b) \( \lim_{x \to 1^+} f(x) \)
This is the limit as \( x \) approaches 1 from the right. Observing the graph:
- As \( x \) approaches 1 from the right, \( f(x) \) is getting closer to 4.
Thus, \( \lim_{x \to 1^+} f(x) = 4 \).

### Part c) \( \lim_{x \to 1} f(x) \)
The overall limit as \( x \to 1 \) exists only if the left-hand limit and right-hand limit are equal. Since:
- \( \lim_{x \to 1^-} f(x) = 2 \)
- \( \lim_{x \to 1^+} f(x) = 4 \)
These are not equal, so the limit does not exist.
Thus, \( \lim_{x \to 1} f(x) \) **does not exist**.

### Part d) \( f(1) \)
This is asking for the value of the function at \( x = 1 \). From the graph:
- At \( x = 1 \), there is a closed circle at \( y = 2 \), which means \( f(1) = 2 \).
Thus, \( f(1) = 2 \).

### Summary of answers:
- a) \( \lim_{x \to 1^-} f(x) = 2 \)
- b) \( \lim_{x \to 1^+} f(x) = 4 \)
- c) \( \lim_{x \to 1} f(x) \) does not exist
- d) \( f(1) = 2 \)

Do you have any questions or would you like more details on these solutions?

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Here are 5 related questions to expand on this:

1. What is the definition of a one-sided limit?
2. How can a limit exist even when the function itself has a jump discontinuity?
3. What is the relationship between left-hand and right-hand limits for a limit to exist?
4. How would you find the limit \( \lim_{x \to 2} f(x) \) from this graph?
5. What happens if \( f(x) \) has a vertical asymptote at \( x = 1 \)?

**Tip:** When analyzing limits, always check both the left-hand and right-hand behavior of the function, especially around points where the function seems discontinuous.

Q2 - Find the following limits. a) lim f(x) as x approaches 1 from the left b) lim f(x) as x approaches 1 from the right c) lim f(x) as x approaches 1 d) f(1) using the graph provided.

Learn how to find one-sided and two-sided limits, including continuity and discontinuity, from a graph. The problem covers concepts such as left-hand and right-hand limits and the conditions for a limit to exist. Suitable for students in Grades 10-12.

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