Let’s analyze the graph to solve each part of the problem.

1. $f(1)$

At $x = 1$, the graph shows that the filled dot (indicating the value of the function) is located at $y = 2$. Thus, $f(1) = 2$.

2. $\lim_{x \to 0} f(x)$

As $x$ approaches 0 from both sides, we observe that the graph approaches the point $y = 2$. Thus, $\lim_{x \to 0} f(x) = 2$.

3. $\lim_{x \to 5^+} f(x)$

This is the limit as $x$ approaches 5 from the right. As we approach $x = 5$ from values greater than 5, the graph appears to approach $y = 3$. Thus, $\lim_{x \to 5^+} f(x) = 3$.

4. $\lim_{x \to 5^-} f(x)$

This is the limit as $x$ approaches 5 from the left. As we approach $x = 5$ from values less than 5, the graph approaches $y = 1$. Thus, $\lim_{x \to 5^-} f(x) = 1$.

Summary of Answers:

• $f(1) = 2$
• $\lim_{x \to 0} f(x) = 2$
• $\lim_{x \to 5^+} f(x) = 3$
• $\lim_{x \to 5^-} f(x) = 1$

Would you like any further details or clarifications on these solutions?

Here are 5 related questions to explore:

1. What is the value of $f(5)$ based on the graph?
2. Does the limit $\lim_{x \to 9} f(x)$ exist?
3. What can you infer if $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$?
4. How would you define continuity at a point based on this graph?
5. How can we use limits to describe behavior at discontinuities in the function?

Tip: When calculating limits, always look carefully at the graph’s behavior from both the left and right to check for any discrepancies or jumps at a point.