The problem you uploaded involves evaluating several statements about limits. Let's break down each option and determine which ones are true based on the properties of limits.

Given:

We have several statements involving the limit of a function $f(x)$ as $x$ approaches $a$ from the left and right.

1. First statement:
   $\lim_{x \to a^+} f(x) = 8 \quad \text{and} \quad \lim_{x \to a^-} f(x) = 8 \quad \text{implies that} \quad \lim_{x \to a} f(x) = 8$
   This is true. If the one-sided limits (from the left and from the right) are both equal to 8, then the two-sided limit exists and is also equal to 8.

2. Second statement:
   $\lim_{x \to a^+} f(x) = 8 \quad \text{implies that} \quad \lim_{x \to a} f(x) = 8$
   This is false. The limit from the right does not necessarily imply the two-sided limit. We need both one-sided limits to be equal to 8 for the two-sided limit to exist and be 8.

3. Third statement:
   $\lim_{x \to a} f(x) = 8 \quad \text{implies that} \quad \lim_{x \to a^+} f(x) = 8$
   This is false. If the two-sided limit exists, we know the limits from both sides must exist and be equal. However, a single-sided limit cannot be inferred from the two-sided limit unless both are considered.

4. Fourth statement:
   $\lim_{x \to a} f(x) = 8 \quad \text{and} \quad \lim_{x \to a^-} f(x) = 8 \quad \text{implies that} \quad \lim_{x \to a^+} f(x) = 8$
   This is true. If the two-sided limit and the left-hand limit are both 8, the right-hand limit must also be 8 for the two-sided limit to exist.

5. Fifth statement:
   $\lim_{x \to a^-} f(x) = 8 \quad \text{implies that} \quad \lim_{x \to a} f(x) = 8$
   This is false. The left-hand limit does not necessarily imply the two-sided limit. Both one-sided limits must be the same for the two-sided limit to exist.

6. Sixth statement:
   $\lim_{x \to a} f(x) = 8 \quad \text{implies that} \quad \lim_{x \to a^-} f(x) = 8$
   This is true. If the two-sided limit is 8, both the left-hand and right-hand limits must also be 8.

Summary of correct answers:

• True: First, Fourth, Sixth
• False: Second, Third, Fifth

Let me know if you need any further clarification or additional details about limits!

Related questions:

1. What is the definition of the two-sided limit?
2. How do we determine if the limit of a function at a point exists?
3. What does it mean for the left-hand limit to equal the right-hand limit?
4. How can a discontinuity in a function affect its limits?
5. How do we approach limits involving infinite values?

Tip: Always check if the left-hand and right-hand limits are equal before concluding that the two-sided limit exists.