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We are tasked with finding the limit of the given function \( f(x) \) as \( x \) approaches 3. The function \( f(x) \) is defined piecewise as follows:

\[
f(x) = 
\begin{cases} 
2 - x^2 & \text{if } x \leq 3, \\
\frac{21}{x} & \text{if } x > 3.
\end{cases}
\]

### Step 1: Evaluate the Left-Hand Limit (\( \lim_{x \to 3^-} f(x) \))
For \( x \leq 3 \), the function is \( f(x) = 2 - x^2 \).

\[
\lim_{x \to 3^-} f(x) = 2 - 3^2 = 2 - 9 = -7.
\]

### Step 2: Evaluate the Right-Hand Limit (\( \lim_{x \to 3^+} f(x) \))
For \( x > 3 \), the function is \( f(x) = \frac{21}{x} \).

\[
\lim_{x \to 3^+} f(x) = \frac{21}{3} = 7.
\]

### Step 3: Determine if the Limit Exists
Since the left-hand limit (\( \lim_{x \to 3^-} f(x) = -7 \)) and the right-hand limit (\( \lim_{x \to 3^+} f(x) = 7 \)) are not equal, the two-sided limit does not exist.

Thus, the correct answer is:

\[
\boxed{D. \text{The limit does not exist}}.
\]

Let me know if you would like further details or have any questions. Here are five related questions for further exploration:

1. What is the definition of a limit for piecewise functions?
2. How do you determine continuity at a point for piecewise functions?
3. How would the limit change if the functions on both sides of the piecewise definition were equal at \( x = 3 \)?
4. What does it mean for a function to have a one-sided limit but no two-sided limit?
5. How does the concept of limits relate to derivatives?

**Tip:** Always check both the left-hand and right-hand limits when dealing with piecewise functions to determine if the overall limit exists.

Given the function f(x) = {2 - x^2 for x ≤ 3, 21/x for x > 3}, find lim(x→3) f(x)

Learn how to find the limit of a piecewise function as x approaches 3. This problem involves evaluating the left-hand and right-hand limits of two different expressions and determining if the overall limit exists. Suitable for students in Grades 10-12.

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