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?

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.