https://file.aiquickdraw.com/mathbot/file/17265929196401bu28f8m.jpg

The image contains a piecewise function \( f(x) \), which is defined as follows:

\[
f(x) = 
\begin{cases} 
x^2 - 9, & x \leq 3 \\
(3 - x)^2, & x > 3
\end{cases}
\]

The text states that the function \( f(x) \) is continuous at \( x = 3 \).

### To check the continuity of \( f(x) \) at \( x = 3 \), we need to confirm the following conditions:
1. The left-hand limit as \( x \) approaches 3 from the left, \( \lim_{x \to 3^-} f(x) \).
2. The right-hand limit as \( x \) approaches 3 from the right, \( \lim_{x \to 3^+} f(x) \).
3. The value of the function at \( x = 3 \), \( f(3) \).

#### Step 1: Left-hand limit
When \( x \leq 3 \), the function is defined as \( f(x) = x^2 - 9 \). So:
\[
\lim_{x \to 3^-} f(x) = 3^2 - 9 = 9 - 9 = 0.
\]

#### Step 2: Right-hand limit
When \( x > 3 \), the function is defined as \( f(x) = (3 - x)^2 \). So:
\[
\lim_{x \to 3^+} f(x) = (3 - 3)^2 = 0^2 = 0.
\]

#### Step 3: Function value at \( x = 3 \)
For \( x = 3 \), we use the first piece of the piecewise function, \( f(x) = x^2 - 9 \), since \( x \leq 3 \):
\[
f(3) = 3^2 - 9 = 9 - 9 = 0.
\]

### Conclusion:
Since the left-hand limit, right-hand limit, and the function value at \( x = 3 \) all equal 0, the function \( f(x) \) is indeed continuous at \( x = 3 \).

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Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How do you check if a function is differentiable at a point?
2. What are the conditions for a function to be continuous over a closed interval?
3. How would the result change if one of the limits did not match the function value at \( x = 3 \)?
4. Can you extend the concept of continuity to a three-variable function?
5. How can we generalize the idea of continuity to piecewise functions in higher dimensions?

**Tip**: Continuity at a point requires both the limits from the left and right to exist and be equal to the function value at that point.

The piecewise function f(x) is defined as: 
 f(x) = 
  x^2 - 9 for x ≤ 3, 
  (3 - x)^2 for x > 3. 
 Determine if the function is continuous at x = 3.

Explore the step-by-step process to determine the continuity of the piecewise function f(x) at x = 3. This example involves calculating left-hand and right-hand limits and evaluating the function's value at x = 3, making it ideal for students studying calculus in Grades 10-12.

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