Let's analyze the graphs in the image to determine the limits as requested.

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

This notation asks for the limit of the function $f(x)$ as $x$ approaches a certain value $c$. Let's go through each graph:

Graph 1:

• Visual Inspection: The graph appears to be a parabola opening downward. As $x$ approaches $c$ from both sides, the graph smoothly approaches a certain value.
• Limit: $\lim_{x \to c} f(x) = L$, where $L$ is the y-value that the curve approaches.

Graph 2:

• Visual Inspection: The graph has a cusp or sharp point at $x = c$. As $x$ approaches $c$ from the left, the graph approaches a different y-value than from the right.
• Limit: The limit does not exist because the left-hand limit ($\lim_{x \to c^-} f(x)$) is not equal to the right-hand limit ($\lim_{x \to c^+} f(x)$).

Graph 3:

• Visual Inspection: The graph appears to be a piecewise function with a jump discontinuity at $x = c$. As $x$ approaches $c$ from the left, it goes to one value, and from the right, it jumps to another.
• Limit: The limit does not exist due to the jump discontinuity.

Graph 4:

• Visual Inspection: The graph shows a function with a hole at $x = c$. The curve approaches the same y-value from both the left and right, but the function is not defined at $x = c$.
• Limit: $\lim_{x \to c} f(x) = L$, where $L$ is the y-value that the graph approaches. The limit exists despite the hole.

Summary:

1. Graph 1: $\lim_{x \to c} f(x)$ exists and equals the y-value of the graph at $x = c$.
2. Graph 2: $\lim_{x \to c} f(x)$ does not exist (DNE) due to the cusp.
3. Graph 3: $\lim_{x \to c} f(x)$ does not exist (DNE) due to the jump discontinuity.
4. Graph 4: $\lim_{x \to c} f(x)$ exists and equals the y-value that the graph approaches despite the hole.

Would you like more detailed explanations for each limit or have any other questions? Here are some related questions to expand your understanding:

1. What are left-hand and right-hand limits?
2. How does the existence of a limit relate to continuity?
3. What is the difference between a removable and a non-removable discontinuity?
4. How can we calculate limits algebraically when a graph is not available?
5. What are common mistakes to avoid when evaluating limits graphically?

Tip: Always consider both the left-hand and right-hand behavior of a function when determining the limit from a graph.