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?

### Given:
- Parking costs Php 10.00 for each hour or part of an hour.
- Let \( C(t) \) represent the cost of parking for \( t \) hours.

#### (a) Sketch of \( C(t) \):

The cost function \( C(t) \) is a **step function** because for each hour (or part of an hour), the cost increases by Php 10. The function can be written as:

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

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

### Sketch:

- \( C(t) \) starts at Php 10 for \( 0 < t \leq 1 \),
- Php 20 for \( 1 < t \leq 2 \),
- Php 30 for \( 2 < t \leq 3 \), and so on.

The graph is made of horizontal line segments with jumps at integer values of \( t \).

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#### (b) Evaluate \( \lim_{t \to t_0^-} C(t) \) and \( \lim_{t \to t_0^+} C(t) \) for \( t_0 > 0 \):

- \( \lim_{t \to t_0^-} C(t) \): This is the **left-hand limit**, which represents the cost as \( t \) approaches \( t_0 \) from the left. Since \( C(t) \) is constant on intervals and jumps at integer values, the left-hand limit at \( t_0 \) is equal to the value of \( C(t) \) just before \( t_0 \).

For example, if \( t_0 = 1 \), the cost just before 1 hour is Php 10.

\[
\lim_{t \to t_0^-} C(t) = 10 \times \lfloor t_0 \rfloor = 10 \times (t_0 - 1)
\]

- \( \lim_{t \to t_0^+} C(t) \): This is the **right-hand limit**, which represents the cost as \( t \) approaches \( t_0 \) from the right. Since parking cost is rounded up, the right-hand limit is higher:

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

Thus, the left-hand and right-hand limits differ by Php 10 at any integer \( t_0 \), leading to a jump.

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

The function \( C(t) \) has a limit at any value of \( t \) **except** at integer values of \( t \). This is because at non-integer values, \( C(t) \) is continuous, but at integer values (like \( t = 1, 2, 3 \)), there is a jump discontinuity where the left-hand limit differs from the right-hand limit by Php 10.

In conclusion, \( C(t) \) has a limit at all **non-integer values** of \( t \), but no limit exists at the integers (e.g., \( t = 1, 2, 3, \dots \)) due to the jump in cost.

---

Do you have any questions or want more details?

#### 5 Related Questions:
1. How does the ceiling function affect the behavior of \( C(t) \)?
2. What other real-life situations can be modeled with a step function like this one?
3. How does the concept of left-hand and right-hand limits apply to continuous functions?
4. What is the formal definition of a discontinuity?
5. Can you calculate the parking cost for a specific time, like 2.5 hours?

#### Tip:
Step functions like this one often arise in real-world scenarios where rates change in discrete intervals, such as billing, shipping, or toll 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 -> t_0^-} C(t) and lim_{t -> t_0^+} C(t).

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

Explore how to sketch the step function C(t) representing parking costs, evaluate left-hand and right-hand limits, and determine the values of t where the function has a limit. This problem involves step functions, limits, and discontinuities, making it suitable for Grades 10-12.

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