The problem asks to evaluate the limit:

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

using the function $f(x)$, which is represented by the graph in the image.

Analysis of the Graph

Looking at the graph:

• As $x$ approaches 1 from the left, $f(x)$ seems to approach the value of 2.
• As $x$ approaches 1 from the right, $f(x)$ also approaches the value of 2.

Since the left-hand limit and right-hand limit are equal, we conclude that:

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

Therefore, the correct answer to the limit is 2.

Would you like further details or explanations on this problem?

Related Questions for Further Exploration:

1. What is the significance of a limit in calculus, and how is it used to analyze function behavior?
2. How would the answer change if the left-hand and right-hand limits were different?
3. How can we determine the continuity of $f(x)$ at $x = 1$ based on the limit?
4. What does it mean if a limit does not exist at a particular point?
5. How can we apply limits to understand real-world phenomena, like rates of change?

Tip:

When determining limits from a graph, always check the behavior of the function as it approaches the point from both the left and the right. This will help in determining if the limit exists and, if so, what the value is.