Let's analyze the given limits involving the function $f(x) = \sqrt{8 - x^3}$ as $x$ approaches 2 from both sides.

1. Left-hand limit $\lim_{x \to 2^-} \sqrt{8 - x^3}$:

As $x \to 2^-$, we substitute values slightly less than 2 into the function $f(x) = \sqrt{8 - x^3}$. When $x = 2$, the expression becomes:

$\sqrt{8 - 2^3} = \sqrt{8 - 8} = 0$

Now, for values of $x$ just less than 2 (like 1.999), $8 - x^3$ will still be slightly positive because $x^3$ will be slightly less than 8, resulting in a positive square root close to 0. Thus:

$\lim_{x \to 2^-} \sqrt{8 - x^3} = 0$

2. Right-hand limit $\lim_{x \to 2^+} \sqrt{8 - x^3}$:

Similarly, as $x \to 2^+$, we substitute values slightly greater than 2 into $f(x) = \sqrt{8 - x^3}$. However, for values of $x$ just greater than 2 (like 2.001), $x^3$ will be slightly greater than 8, making $8 - x^3$ negative.

Since the square root of a negative number is not defined in the real number system, the function $f(x) = \sqrt{8 - x^3}$ becomes undefined as $x \to 2^+$.

Thus:

$\lim_{x \to 2^+} \sqrt{8 - x^3} \quad \text{is undefined}.$

3. Overall limit $\lim_{x \to 2} \sqrt{8 - x^3}$:

Since the right-hand limit does not exist in the real number system, the overall limit does not exist. A limit only exists if both the left-hand and right-hand limits exist and are equal.

Conclusion:

• $\lim_{x \to 2^-} \sqrt{8 - x^3} = 0$
• $\lim_{x \to 2^+} \sqrt{8 - x^3}$ does not exist (undefined)
• Therefore, the overall limit $\lim_{x \to 2} \sqrt{8 - x^3}$ does not exist.

Hence, the correct answer is:

$\lim_{x \to 2^-} \sqrt{8 - x^3} = 0$

Would you like more details on the process? Here are some additional related questions for practice:

1. What happens to the function $f(x) = \sqrt{8 - x^3}$ as $x \to 3$?
2. How do we handle square roots of negative values in complex numbers?
3. What is the definition of a two-sided limit?
4. What does it mean for a function to be undefined at a point?
5. How do we compute limits involving piecewise functions?

Tip: Always check both the left-hand and right-hand limits when dealing with limits to ensure they match for the overall limit to exist.