1. **Problem Statement:** Given that $$\lim_{x \to 3} f(x) = 7$$, determine which of the following statements must be true: I. $$f$$ is continuous at $$x=3$$ II. $$f$$ is differentiable at $$x=3$$ III. $$f(3) = 7$$ Options: A) None B) II only C) III only D) I and III only E) I, II, and III 2. **Recall the definitions and rules:** - The limit $$\lim_{x \to a} f(x) = L$$ means as $$x$$ approaches $$a$$, $$f(x)$$ approaches $$L$$. - $$f$$ is continuous at $$x=a$$ if and only if: $$\lim_{x \to a} f(x) = f(a)$$ - Differentiability at $$x=a$$ implies continuity at $$x=a$$, but continuity does not imply differentiability. 3. **Analyze each statement:** - Statement I (Continuity): For $$f$$ to be continuous at $$x=3$$, we need $$\lim_{x \to 3} f(x) = f(3)$$. - Statement III: $$f(3) = 7$$ is not guaranteed by the limit alone; the function value at 3 could be different or undefined. - Statement II (Differentiability): Differentiability requires continuity and more (smoothness), so it is not guaranteed by the limit alone. 4. **Conclusion:** - Since $$\lim_{x \to 3} f(x) = 7$$, the limit exists and equals 7. - But $$f(3)$$ may or may not equal 7, so continuity (I) and $$f(3) = 7$$ (III) are not guaranteed. - Differentiability (II) is also not guaranteed. Therefore, none of the statements must be true based solely on the given limit. **Final answer:** A) None