Based on your description of the graph, here’s how we can analyze the behavior of the function $f$ at $x = 4$.

1. Curve behavior near $x = 4$:

   • There is an open circle at $(4, 1)$, meaning the curve approaches $y = 1$ from both sides but does not include this point.
   • A closed circle is located at $(4, -4)$, which means the function is defined at $x = 4$ with $f(4) = -4$.

2. Analyzing the limits:

   • $\lim_{x \to 4^-} f(x) = 1$, since the curve approaches $y = 1$ from the left side.
   • $\lim_{x \to 4^+} f(x) = 1$, since the curve also approaches $y = 1$ from the right side.
   • Both left-hand and right-hand limits exist and are equal to 1.

3. Does $\lim_{x \to 4} f(x)$ exist?

   • Yes, because both the left-hand and right-hand limits exist and are equal to 1. So,
     $\lim_{x \to 4} f(x) = 1$.

4. Is the function defined at $x = 4$?

   • Yes, since there is a closed circle at $(4, -4)$, the function is defined at $x = 4$, and $f(4) = -4$.

5. Is the function continuous at $x = 4$?

   • No, because for continuity, the limit $\lim_{x \to 4} f(x)$ must equal the function value $f(4)$. In this case: $\lim_{x \to 4} f(x) = 1 \quad \text{and} \quad f(4) = -4$ Since these are not equal, $f$ is not continuous at $x = 4$.

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Correct Answers:

• A: Both $\lim_{x \to 4^+} f(x)$ and $\lim_{x \to 4^-} f(x)$ exist.
• B: $\lim_{x \to 4} f(x)$ exists.
• C: $f$ is defined at $x = 4$.

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Would you like more clarification or further details?

Here are 5 related questions:

1. How do we determine left-hand and right-hand limits for other points?
2. What are the conditions for a function to be continuous at a point?
3. How do open and closed circles affect the definition of a function?
4. What happens if the left-hand and right-hand limits are not equal?
5. How can discontinuities be classified (removable, jump, infinite)?

Tip: Continuity requires that the function value, left-hand limit, and right-hand limit all match at a given point.