which of the following is true: lim x 2- (8-x^3)^1/2 =0 limx2+ (8-x^3)^1/2=0 lim x 2 (8-x^3)^1/2 =0 or lim x 2- (8-x^3)^1/2 = limx2+ (8-x^3)^1/2=0

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.

Explore the one-sided limits of the function √(8 - x^3) as x approaches 2. This problem focuses on limit analysis, understanding undefined limits, and square root functions in real numbers. Ideal for Grades 11-12 students mastering calculus concepts.

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