Based on the provided image, you need to determine the limits for the given functions as they approach specific values of $x$. Let’s analyze each graph to find the limits.

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

• Interpretation: This represents the limit of the function $f(x)$ as $x$ approaches $a$ from the left.
• Graph: The function appears to approach a certain value as $x$ gets close to $a$ from the left. From the graph, it looks like as $x$ approaches $a$ from the left, the function value approaches a certain $y$-value.
• Result: $\lim_{x \to a^-} f(x) = L$, where $L$ is the $y$-value the function approaches.

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

• Interpretation: This represents the limit of the function $f(x)$ as $x$ approaches $a$ from the right.
• Graph: The graph shows that as $x$ approaches $a$ from the right, the function approaches a different $y$-value.
• Result: $\lim_{x \to a^+} f(x) = M$, where $M$ is the $y$-value the function approaches.

3. $\lim_{x \to a} f(x)$

• Interpretation: This represents the limit of the function $f(x)$ as $x$ approaches $a$ from both sides.
• Graph: The limit exists if the left-hand limit and right-hand limit are equal. If they are not equal, the limit does not exist.
• Result:
  • If $L = M$, then $\lim_{x \to a} f(x) = L = M$.
  • If $L \neq M$, then $\lim_{x \to a} f(x)$ does not exist.

4. $\lim_{x \to b} f(x)$

• Interpretation: This is similar to the previous problem, where you must determine the limit as $x$ approaches $b$.
• Graph: Examine the graph near $x = b$ and check if the function approaches the same value from both the left and the right.
• Result: Again, if the left-hand limit equals the right-hand limit, the limit exists and is equal to this common value.

5. $\lim_{x \to c} f(x)$

• Interpretation: This is the limit as $x$ approaches $c$.
• Graph: Like the previous cases, analyze if the function approaches the same value from both directions near $x = c$.
• Result: Determine whether the left and right-hand limits are equal or not, and conclude whether the limit exists.

If you provide specific points $a$, $b$, and $c$, I can help with exact limit values.

Would you like more detailed explanations or have any questions? Here are five relative questions to expand this concept:

1. What happens to the limit if the graph has a jump discontinuity?
2. How do you determine the limit if the function is undefined at the point but approaches a value?
3. What is the significance of one-sided limits in determining the overall limit?
4. How would you find the limit if the function has an infinite discontinuity?
5. How can you graphically confirm that a limit does not exist?

Tip: Always check both the left-hand and right-hand limits separately before concluding the overall limit at a point.