Let's solve the problem using the graph provided:

1. Finding $f(1)$:

From the graph, we observe the point where $x = 1$. The graph shows a hole (an open circle) at $x = 1$, and the solid point above it is located at $y = 1$. This point indicates that $f(1) = 1$.

Answer: $f(1) = 1$

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

To find the limit of $f(x)$ as $x$ approaches 1, we need to consider the behavior of the function from both the left and the right sides of $x = 1$.

• From the left ($x \to 1^-$), the function approaches $y = 0$.
• From the right ($x \to 1^+$), the function also approaches $y = 0$.

Since both sides approach the same value, the limit exists.

Answer: $\lim_{{x \to 1}} f(x) = 0$

Would you like further details on these calculations, or do you have any other questions?

Here are some related questions you might find interesting:

1. What happens to $f(x)$ as $x \to 0$?
2. What is the limit of $f(x)$ as $x \to -1$?
3. Does the function $f(x)$ have any asymptotes?
4. How would you describe the continuity of $f(x)$ at $x = 1$?
5. Can you determine $f(x)$ at $x = 2$?

Tip: When finding limits graphically, always observe the function's behavior from both sides of the point of interest to ensure the limit exists and is the same from both directions.