To solve this problem, we need to determine the limit of the function $f(x)$ as $x$ approaches 4, using the graph provided.

Step-by-step Solution:

1. Identify the left-hand limit as $x \to 4^-$:

   • As $x$ approaches 4 from the left, the values of $f(x)$ approach $-2$.

2. Identify the right-hand limit as $x \to 4^+$:

   • As $x$ approaches 4 from the right, the values of $f(x)$ approach 2.

3. Check if the left-hand limit equals the right-hand limit:

   • The left-hand limit is $-2$ and the right-hand limit is $2$.
   • Since $-2 \neq 2$, the left-hand limit and right-hand limit are not equal.

4. Conclusion:

   • The limit does not exist because the left-hand and right-hand limits are not equal.

Answer:

The correct answer is: "does not exist because the left- and right-hand limits are not equal."

Would you like more details on limits or have any further questions?

5 Related Questions:

1. What is the definition of a limit in calculus?
2. How do you determine the left-hand and right-hand limits from a graph?
3. What does it mean for a function to be continuous at a point?
4. How do limits relate to the concept of derivatives?
5. What is the difference between one-sided limits and two-sided limits?

Tip:

When analyzing limits on a graph, always check both the left-hand and right-hand limits to see if they converge to the same value. If they do, the limit exists; if they don't, the limit does not exist.