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In this problem, we are given that the function \( f \) is continuous at \( x = 2 \), and we know the following:

- \( \lim\limits_{x \to 2^-} f(x) = 4 \) (the left-hand limit as \( x \) approaches 2)
- \( \lim\limits_{x \to 2^+} f(x) = 4 \) (the right-hand limit as \( x \) approaches 2)

Since the function is continuous at \( x = 2 \), the value of the function at \( x = 2 \) must be equal to the limit of the function as \( x \) approaches 2 from both sides. In other words:

\[
f(2) = \lim\limits_{x \to 2} f(x)
\]

Given that both the left-hand and right-hand limits are equal to 4, we conclude that:

\[
f(2) = 4
\]

Thus, the correct answer is:

\[
f(2) = 4
\]

Would you like further details or have any questions about this solution?

Here are some follow-up questions for deeper understanding:

1. What does it mean for a function to be continuous at a point?
2. What is the difference between one-sided limits and a two-sided limit?
3. Can a function have different left-hand and right-hand limits at a point and still be continuous?
4. How can you use limits to identify discontinuities in a function?
5. How would you define a removable discontinuity?

**Tip:** A function is continuous at a point if its value at that point equals the limit as you approach the point from both directions.

For a function f that is continuous at x = 2, with lim f(x) = 4 as x approaches 2 from the left, and lim f(x) = 4 as x approaches 2 from the right, find f(2).

Explore how to solve a continuity problem where the function is continuous at x = 2. Learn about limits and one-sided limits to find f(2) when lim(x->2) f(x) = 4 from both sides. Suitable for Grades 10-12.

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