To check whether the function $f(x)$ is continuous at $x = 2$, we need to verify the following:

1. Left-hand limit: The limit of $f(x)$ as $x \to 2^-$ (from the left).
2. Right-hand limit: The limit of $f(x)$ as $x \to 2^+$ (from the right).
3. Value of the function at $x = 2$: We are given $f(2) = 1$.

Left-hand limit:

For $x \leq 2$, the function is given by:

$f(x) = \frac{4}{x + 2}$

We want to find the left-hand limit as $x \to 2^-$:

$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} \frac{4}{x + 2} = \frac{4}{2 + 2} = \frac{4}{4} = 1$

Right-hand limit:

For $x > 2$, the function is given by:

$f(x) = \frac{x^2 - 4}{x - 2}$

This expression simplifies as follows (factor the numerator):

$f(x) = \frac{(x - 2)(x + 2)}{x - 2}$

Canceling out the $(x - 2)$ terms:

$f(x) = x + 2 \quad \text{for} \quad x > 2$

Now, compute the right-hand limit as $x \to 2^+$:

$\lim_{x \to 2^+} f(x) = 2 + 2 = 4$

Conclusion:

• Left-hand limit $\lim_{x \to 2^-} f(x) = 1$
• Right-hand limit $\lim_{x \to 2^+} f(x) = 4$
• $f(2) = 1$

Since the left-hand limit and the right-hand limit are not equal (1 ≠ 4), the function is not continuous at $x = 2$.

Thus, the answer is False (F). The function is not continuous.

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Would you like to explore further details on continuity, limits, or anything else related? Here are some related questions:

1. What is the general definition of continuity at a point?
2. How can you determine if a function is continuous from its graph?
3. What happens when the left and right limits of a function at a point are different?
4. How do you calculate limits for rational functions?
5. Can you explain how to handle indeterminate forms in limits?

Tip: Remember, for a function to be continuous at a point, the left-hand limit, right-hand limit, and function value at that point must all be equal.