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Let’s go through each statement one by one and analyze whether each is true or false.

1. **If** \( \lim_{x \to 5} f(x) = 6 \) **and** \( \lim_{x \to 5} g(x) = 0 \), **then** \( \lim_{x \to 5} \left( \frac{f(x)}{g(x)} \right) \) **does not exist.**

   - When the limit of the denominator \( g(x) \) is 0 while the limit of the numerator \( f(x) \) is non-zero (here, 6), the limit \( \lim_{x \to 5} \left( \frac{f(x)}{g(x)} \right) \) typically does not exist due to division by zero.
   - **Answer: True**

2. **If** \( f'(6) \) **exists, then the limit** \( \lim_{x \to 6} f(x) \) **is** \( f(6) \).

   - The existence of \( f'(6) \) implies that \( f(x) \) is continuous at \( x = 6 \), meaning \( \lim_{x \to 6} f(x) = f(6) \).
   - **Answer: True**

3. **If** \( f(x) \) **is differentiable at** \( a \), **then** \( f(x) \) **is continuous at** \( a \).

   - Differentiability at a point implies continuity at that point. If \( f(x) \) is differentiable at \( a \), then \( f(x) \) must also be continuous at \( a \).
   - **Answer: True**

4. **If** \( p(x) \) **is a polynomial, then** \( \lim_{x \to 6} p(x) \) **is** \( p(6) \).

   - Polynomials are continuous everywhere. Therefore, \( \lim_{x \to 6} p(x) = p(6) \).
   - **Answer: True**

5. **If** \( f(x) \) **is continuous at** \( a \), **then** \( f(x) \) **is differentiable at** \( a \).

   - Continuity at a point does not guarantee differentiability. A function can be continuous but not differentiable (e.g., \( f(x) = |x| \) at \( x = 0 \)).
   - **Answer: False**

### Summary of Answers:
1. True  
2. True  
3. True  
4. True  
5. False  

Would you like more details on any of these answers or have any other questions?

### Related Questions to Expand Understanding:
1. What are the conditions required for differentiability at a point?
2. Can you provide an example of a function that is continuous but not differentiable at a certain point?
3. How does the Squeeze Theorem relate to limits involving division by zero?
4. Why does differentiability imply continuity, but not vice versa?
5. What are some properties of polynomial functions regarding limits and continuity?

### Tip:
When evaluating limits involving fractions, always check if the limit of the denominator is zero, as this can indicate a potential for non-existence or the need for further analysis, such as applying L'Hôpital's Rule.

Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false.

Answer true or false questions about limits, continuity, and differentiability. Understand core calculus concepts, including limits, continuity, differentiability, and polynomial behavior.

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