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Find the number of the correct statements.

To determine the number of correct statements regarding the continuity of the functions \( p(t) \) and \( q(t) \), we need to analyze each function at the points where their definitions change: \( t = 3 \) for \( p(t) \) and \( t = 2 \) for \( q(t) \).

### Function \( p(t) \)

The function \( p(t) \) is defined as:
\[ p(t) = \begin{cases} 
\frac{t^3 - 27}{t - 3} & \text{if } 0 \leq t < 3, \\
27 & \text{if } t = 3, \\
\frac{1}{e^{81} (t-3)} (e^{27t} - e^{81}) & \text{if } t > 3.
\end{cases} \]

#### Checking Continuity at \( t = 3 \)

1. **Left-hand limit as \( t \to 3^- \)**:
\[ \lim_{t \to 3^-} p(t) = \lim_{t \to 3^-} \frac{t^3 - 27}{t - 3} \]
   Using the factorization \( t^3 - 27 = (t-3)(t^2 + 3t + 9) \):
\[ \lim_{t \to 3^-} \frac{(t-3)(t^2 + 3t + 9)}{t-3} = \lim_{t \to 3^-} (t^2 + 3t + 9) = 27 \]

2. **Value at \( t = 3 \)**:
\[ p(3) = 27 \]

3. **Right-hand limit as \( t \to 3^+ \)**:
\[ \lim_{t \to 3^+} p(t) = \lim_{t \to 3^+} \frac{1}{e^{81} (t-3)} (e^{27t} - e^{81}) \]
   As \( t \to 3^+ \), \( e^{27t} \to e^{81} \), so:
\[ \lim_{t \to 3^+} p(t) = \lim_{t \to 3^+} \frac{1}{e^{81} (t-3)} (e^{27t} - e^{81}) = \lim_{t \to 3^+} \frac{e^{81}(e^{27(t-3)} - 1)}{e^{81} (t-3)} = \lim_{t \to 3^+} \frac{e^{27(t-3)} - 1}{t-3} \]
   Using the approximation \( e^x \approx 1 + x \) for small \( x \):
\[ = \lim_{t \to 3^+} 27 = 27 \]

Since the left-hand limit, the value of the function at \( t = 3 \), and the right-hand limit all equal 27, \( p(t) \) is continuous at \( t = 3 \).

### Function \( q(t) \)

The function \( q(t) \) is defined as:
\[ q(t) = \begin{cases} 
(5t - 9)^{\frac{1}{t-2}} & \text{if } 0 \leq t < 2, \\
e^4 & \text{if } t = 2, \\
\frac{e^{t^2 - e^4}}{t - 2} & \text{if } t > 2.
\end{cases} \]

#### Checking Continuity at \( t = 2 \)

1. **Left-hand limit as \( t \to 2^- \)**:
\[ \lim_{t \to 2^-} q(t) = \lim_{t \to 2^-} (5t - 9)^{\frac{1}{t-2}} \]
   As \( t \to 2^- \), \( 5t - 9 \to 1 \), and \( \frac{1}{t-2} \to \infty \). Hence, \( (5t - 9)^{\frac{1}{t-2}} \to 1^+ \to 1 \).

2. **Value at \( t = 2 \)**:
\[ q(2) = e^4 \]

3. **Right-hand limit as \( t \to 2^+ \)**:
\[ \lim_{t \to 2^+} q(t) = \lim_{t \to 2^+} \frac{e^{t^2 - e^4}}{t - 2} \]
   As \( t \to 2^+ \), the term \( \frac{e^{t^2 - e^4}}{t - 2} \to \infty \) unless \( e^{t^2 - e^4} = 0 \), which it does not.

Thus, \( q(t) \) is not continuous at \( t = 2 \).

### Summary

- \( p(t) \) is continuous.
- \( q(t) \) is not continuous.

So, the correct statements are:

- Statement Q: \( p(t) \) is continuous, but \( q(t) \) is not.

There is 1 correct statement.

### Additional Questions
1. What is the left-hand limit of \( p(t) \) as \( t \) approaches 3?
2. What is the right-hand limit of \( p(t) \) as \( t \) approaches 3?
3. What is the left-hand limit of \( q(t) \) as \( t \) approaches 2?
4. What is the right-hand limit of \( q(t) \) as \( t \) approaches 2?
5. Why is \( q(t) \) not continuous at \( t = 2 \)?
6. What is the value of \( p(t) \) at \( t = 3 \)?
7. How can you approximate \( e^x \) for small \( x \)?
8. What are the conditions for a function to be continuous at a point?

### Tip

Always check the limits from both sides and the value of the function at the point to determine continuity.

Explore the continuity analysis of functions p(t) and q(t) at specific points. Understand the concept of continuity, limits, and piecewise functions in this detailed mathematical analysis. Suitable for high school students studying calculus and advanced algebra.

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