From the graph shown, we are asked to determine:

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

Analyzing the Graph:

• As $x \to 2$ from the left ($x \to 2^{-}$):
  • The graph approaches $y = 3$.
• As $x \to 2$ from the right ($x \to 2^{+}$):
  • The graph approaches $y = 1$.

Conclusion:

Since the left-hand limit and the right-hand limit as $x \to 2$ are different, the overall limit does not exist.

$\lim_{x \to 2} f(x) \quad \text{does not exist}.$

Would you like a more detailed explanation on limits? Here are some additional questions that could expand your understanding:

1. What are the conditions for the existence of a limit?
2. How can we calculate one-sided limits?
3. What is the difference between limits and continuity at a point?
4. How do discontinuities like jumps or holes affect limits?
5. Can a limit exist if the function is undefined at the point?

Tip: When both the left-hand and right-hand limits differ, the two-sided limit does not exist.