If we know for a function f(x)f(x) that f(3)=5f(3)=5, which of the following must be true in order for f(x)f(x) to be continuous at x=3x=3?

**I.** limx→3+f(x)=limx→3−f(x)limx→3+⁡f(x)=limx→3−⁡f(x)
**II.** limx→3f(x)=5limx→3⁡f(x)=5
**III.** limx→5f(x)=3

To determine which statements must be true for the function \(f(x)\) to be continuous at \(x = 3\), let's first recall the definition of continuity at a point. A function \(f(x)\) is continuous at \(x = a\) if the following three conditions are met:

1. The function is defined at \(x = a\), i.e., \(f(a)\) exists.
2. The limit of the function as \(x\) approaches \(a\) exists, i.e., \(\lim_{x \to a} f(x)\) exists.
3. The limit of the function as \(x\) approaches \(a\) is equal to the function's value at that point, i.e., \(\lim_{x \to a} f(x) = f(a)\).

Given in the problem:

- \(f(3) = 5\)

To ensure continuity at \(x = 3\), let's analyze each statement:

### Statement I: \(\lim_{x \to 3^+} f(x) = \lim_{x \to 3^-} f(x)\)

- **Explanation:** This statement asserts that the left-hand limit (\(\lim_{x \to 3^-} f(x)\)) and the right-hand limit (\(\lim_{x \to 3^+} f(x)\)) of \(f(x)\) as \(x\) approaches 3 are equal.
- **Necessity for Continuity:** This is necessary for the overall limit \(\lim_{x \to 3} f(x)\) to exist. If the left-hand and right-hand limits are not equal, the limit \(\lim_{x \to 3} f(x)\) does not exist.
- **Conclusion:** **True.** This must be true for continuity at \(x = 3\).

### Statement II: \(\lim_{x \to 3} f(x) = 5\)

- **Explanation:** This statement asserts that the limit of \(f(x)\) as \(x\) approaches 3 is equal to 5.
- **Necessity for Continuity:** Since \(f(3) = 5\), for \(f(x)\) to be continuous at \(x = 3\), it must also be true that \(\lim_{x \to 3} f(x) = f(3) = 5\).
- **Conclusion:** **True.** This must be true for continuity at \(x = 3\).

### Statement III: \(\lim_{x \to 5} f(x) = 3\)

- **Explanation:** This statement asserts that the limit of \(f(x)\) as \(x\) approaches 5 is equal to 3.
- **Necessity for Continuity:** This statement is irrelevant to the continuity of \(f(x)\) at \(x = 3\). Continuity at \(x = 3\) does not depend on the behavior of the function at \(x = 5\).
- **Conclusion:** **False.** This is not required for continuity at \(x = 3\).

### Final Answer

The statements that **must be true** for \(f(x)\) to be continuous at \(x = 3\) are:

\[
\text{I and II only.}
\]

### Conclusion:

\[
\boxed{\text{I and II}}
\]

Do you have any questions, or would you like further details? Here are some related questions to explore:

1. How can we determine if a function is continuous over an interval?
2. What are the different types of discontinuities in functions?
3. How does differentiability relate to continuity?
4. What is the difference between one-sided and two-sided limits?
5. How can a piecewise function be made continuous at a point of interest?

**Tip:** For a function to be continuous at a point, check if the limit from both sides matches the function value at that point!

Explore the conditions required for a function f(x) to be continuous at x = 3. Analyze the statements regarding limits and function values to determine continuity. Suitable for college-level students studying calculus and mathematical analysis.

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