Parking in a local supermarket costs Php 10.00 for each hour or part of an hour. Let  C(t)  be the function representing the cost of parking for  t  hours.

(a) Sketch  C(t) .

(b) For an arbitrary number  t_0 > 0 , evaluate  \lim_{t \to t_0^-} C(t)  and  \lim_{t \to t_0^+} C(t) .

(c) At what values of  t  does  C(t)  have a limit?

Let's break this problem down step by step:

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### (a) **Sketching \( C(t) \):**

The cost of parking \( C(t) \) is given as Php 10.00 for each hour or part of an hour. This implies that \( C(t) \) is a step function, where the cost jumps by Php 10.00 after each hour. We can define the function mathematically as:

\[
C(t) = 10 \times \lceil t \rceil
\]

where \( \lceil t \rceil \) is the ceiling function, which rounds \( t \) up to the nearest integer.

- For \( 0 < t \leq 1 \), the cost is Php 10.00.
- For \( 1 < t \leq 2 \), the cost is Php 20.00.
- For \( 2 < t \leq 3 \), the cost is Php 30.00.
- And so on.

This creates a stepwise increasing graph where each interval is open on the left and closed on the right, representing that the cost jumps at each hour.

---

### (b) **Evaluating \( \lim_{t \to t_0^-} C(t) \) and \( \lim_{t \to t_0^+} C(t) \):**

For any arbitrary \( t_0 > 0 \), let's evaluate the one-sided limits:

1. **Left-hand limit \( \lim_{t \to t_0^-} C(t) \):**
   - This limit considers the values of \( t \) approaching \( t_0 \) from the left. As we approach \( t_0 \) from values smaller than \( t_0 \), the cost function \( C(t) \) will be equal to \( 10 \times \lceil t_0 - 1 \rceil \) (since \( t \) is not yet rounded up to \( t_0 \)).
   
   Therefore:
   \[
   \lim_{t \to t_0^-} C(t) = 10 \times \lceil t_0 - 1 \rceil = 10 \times \lfloor t_0 \rfloor
   \]

2. **Right-hand limit \( \lim_{t \to t_0^+} C(t) \):**
   - This limit considers the values of \( t \) approaching \( t_0 \) from the right. As soon as \( t \) exceeds \( t_0 \), the ceiling function rounds \( t \) up to \( t_0 + 1 \), meaning the cost increases to the next multiple of 10.

   Therefore:
   \[
   \lim_{t \to t_0^+} C(t) = 10 \times \lceil t_0 \rceil
   \]

Since the step function jumps at each integer value, the left-hand limit will always be less than the right-hand limit at integer values.

---

### (c) **At what values of \( t \) does \( C(t) \) have a limit?**

The function \( C(t) \) will only have a limit where the left-hand and right-hand limits are equal. This can only happen at non-integer values of \( t \), since at integer values the function experiences a jump (discontinuity).

- **For non-integer values of \( t \):**
   - Both the left-hand limit and the right-hand limit will approach the same value, so \( C(t) \) will have a limit at these points.

- **For integer values of \( t \):**
   - The left-hand and right-hand limits differ (since the cost jumps by Php 10.00 at each hour), so \( C(t) \) does not have a limit at these points.

In summary, \( C(t) \) has a limit at all non-integer values of \( t \), but not at integer values where the function has discontinuities.

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### Follow-up Questions:

1. How would the graph change if the parking rate increased by Php 5.00 for each additional hour?
2. Can you find an alternative function to model this cost that removes the discontinuities?
3. How does the ceiling function impact the nature of the cost function?
4. What would happen to the limits if the cost was Php 10.00 for each *half-hour* instead of each full hour?
5. Can you explain why the left-hand and right-hand limits differ at integer values?

**Tip:** The ceiling function is essential for modeling scenarios where rounding up happens in practical applications like parking charges or billing systems.

Parking in a local supermarket costs Php 10.00 for each hour or part of an hour. Let C(t) be the function representing the cost of parking for t hours.

(a) Sketch C(t).

(b) For an arbitrary number t_0 > 0, evaluate \lim_{t \to t_0^-} C(t) and \lim_{t \to t_0^+} C(t).

(c) At what values of t does C(t) have a limit?

Explore how to model parking costs with a step function where the price increases by Php 10.00 per hour. This problem focuses on the ceiling function, evaluating one-sided limits, and identifying where the function is discontinuous. Suitable for Grades 11-12.

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