From the graph you uploaded, we need to determine the limits:

1. $\lim_{x \to 2^-} f(x)$
2. $\lim_{x \to 2^+} f(x)$

Analyzing the Graph:

• The graph has two points of interest marked as $P$ and $Q$.
• As $x$ approaches 2 from the left ($2^-$), the $y$-value of $P$ appears to be the relevant limit.
• As $x$ approaches 2 from the right ($2^+$), the $y$-value of $Q$ appears to be the relevant limit.

Observations:

• Point $P$ has coordinates $(2, 4)$.
• Point $Q$ has coordinates $(2, -2)$.

Calculating the Limits:

1. Left-hand limit as $x$ approaches 2: $\lim_{x \to 2^-} f(x) = 4$
2. Right-hand limit as $x$ approaches 2: $\lim_{x \to 2^+} f(x) = -2$

These limits indicate a jump discontinuity at $x = 2$, as the left-hand and right-hand limits are not equal.

Would you like to explore further details or have any other questions about this analysis?

Related Questions:

1. Can you describe what a jump discontinuity is?
2. Why is it important to check both left-hand and right-hand limits?
3. How would you evaluate a limit at a point where there is a hole in the graph?
4. What happens to the overall continuity of a function if there are jump discontinuities?
5. How would the function’s behavior change if the points $P$ and $Q$ were connected with a line?

Tip: When analyzing limits from a graph, it’s crucial to carefully observe where the function is approaching as $x$ moves towards the point of interest from different directions.