Let's analyze the graph and solve the given questions:

1. $f(1)$:

We need to find the value of the function $f(x)$ at $x = 1$.

Looking at the graph, there is a filled dot at $(1, 2)$, indicating that the value of $f(1) = 2$. So: $f(1) = 2$

2. $\lim_{x \to 1} f(x)$:

Now we calculate the limit of $f(x)$ as $x$ approaches 1. This requires checking both the left-hand limit ($x \to 1^-$) and the right-hand limit ($x \to 1^+$).

• As $x \to 1^-$, the graph approaches 3 (from the left).
• As $x \to 1^+$, the graph approaches 3 (from the right).

Since both limits agree, we conclude: $\lim_{x \to 1} f(x) = 3$

Summary:

• $f(1) = 2$
• $\lim_{x \to 1} f(x) = 3$

Would you like further details or clarifications?

Here are 5 related questions to further explore these ideas:

1. What happens if the limit from the left and right do not match?
2. Can a function be discontinuous at a point and still have a limit at that point?
3. How do you determine if a function is continuous at a certain point?
4. What are the conditions for the existence of a limit?
5. How do limits apply to real-world situations like speed or temperature changes?

Tip: Always check both one-sided limits when evaluating limits to ensure they match, confirming the existence of the overall limit.