The Hourly Parking Garage at BWI International Airport in Baltimore costs $2 per half hour during the first hour and $4 for each hour or part of an hour for more than 1 hour and up to 5 hours. If 

*C* = *C*(*t*)

 is the total charge for *t* hours in BWI's Hourly Parking Garage, write a piecewise function that describes 

*C*(*t*)

 for 

*t* < 3.

 Use 

*C*(*t*)

 to find the following limits, if they exist. (If an answer does not exist, enter DNE.)

(a)    

lim *t*→0.5− *C*(*t*)


$  

(b)    

lim *t*→0.5+ *C*(*t*)


$  

(c)    

lim *t*→0.5 *C*(*t*)


$  

(d)    

lim *t*→2.5 *C*(*t*)


$  

Let's break down the problem and construct the piecewise function $ C(t) $ for the total charge $ C $ as a function of time $ t $ for $ t < 3 $ hours, based on the parking rates provided.

### Piecewise Function for $ C(t) $:
1. **For the first hour ( $ 0 < t \leq 1 $ ):**
   - The charge is $2 for each half hour. This means:
     - For $ 0 < t \leq 0.5 $, $ C(t) = 2 $
     - For $ 0.5 < t \leq 1 $, $ C(t) = 4 $

2. **For more than 1 hour ( $ 1 < t \leq 3 $ ):**
   - After 1 hour, the charge becomes $4 for each additional hour or part of an hour.
     - For $ 1 < t \leq 2 $, $ C(t) = 8 $ (since 1 full hour adds $4 to the previous $4 charge)
     - For $ 2 < t \leq 3 $, $ C(t) = 12 $

Thus, the piecewise function $ C(t) $ for $ t < 3 $ is:
\[
C(t) = 
\begin{cases} 
2 & \text{if } 0 < t \leq 0.5 \\
4 & \text{if } 0.5 < t \leq 1 \\
8 & \text{if } 1 < t \leq 2 \\
12 & \text{if } 2 < t \leq 3
\end{cases}
\]

### Now, let's calculate the limits.

---

**(a) $\lim_{t \to 0.5^-} C(t)$:**

- As $ t $ approaches 0.5 from the left ( $ t < 0.5 $ ), the charge is $2. So,
  \[
  \lim_{t \to 0.5^-} C(t) = 2
  \]

---

**(b) $\lim_{t \to 0.5^+} C(t)$:**

- As $ t $ approaches 0.5 from the right ( $ t > 0.5 $ ), the charge is $4. So,
  \[
  \lim_{t \to 0.5^+} C(t) = 4
  \]

---

**(c) $\lim_{t \to 0.5} C(t)$:**

- Since the left-hand limit $ \lim_{t \to 0.5^-} C(t) = 2 $ and the right-hand limit $ \lim_{t \to 0.5^+} C(t) = 4 $ are not equal, the two-sided limit does not exist. So,
  \[
  \lim_{t \to 0.5} C(t) = \text{DNE}
  \]

---

**(d) $\lim_{t \to 2.5} C(t)$:**

- As $ t $ approaches 2.5 from either direction, the charge remains $12. So,
  \[
  \lim_{t \to 2.5} C(t) = 12
  \]

---

### Summary:
(a) $ \lim_{t \to 0.5^-} C(t) = 2 $

(b) $ \lim_{t \to 0.5^+} C(t) = 4 $

(c) $ \lim_{t \to 0.5} C(t) = \text{DNE} $

(d) $ \lim_{t \to 2.5} C(t) = 12 $

---

Would you like further explanation or examples related to limits or piecewise functions? Here are 5 follow-up questions to expand on the topic:

1. How would the piecewise function $ C(t) $ change for $ t > 5 $?
2. Can we find the derivative of the piecewise function $ C(t) $?
3. What is the significance of left-hand and right-hand limits in real-world problems?
4. How do we calculate limits when the function is not piecewise but continuous?
5. What happens to the function $ C(t) $ if there's a discount applied after a certain number of hours?

**Tip:** For limits that don't exist due to discontinuities, always check both the left-hand and right-hand limits to see if they converge to the same value.

The Hourly Parking Garage at BWI International Airport in Baltimore costs $2 per half hour during the first hour and $4 for each hour or part of an hour for more than 1 hour and up to 5 hours. If C = C(t) is the total charge for t hours in BWI's Hourly Parking Garage, write a piecewise function that describes C(t) for t < 3. Use C(t) to find the following limits, if they exist. (If an answer does not exist, enter DNE.)

(a) lim t→0.5− C(t)

(b) lim t→0.5+ C(t)

(c) lim t→0.5 C(t)

(d) lim t→2.5 C(t)


Explore a detailed breakdown of parking charges at BWI's Hourly Parking Garage using a piecewise function for t < 3 hours. Learn how to calculate limits and identify discontinuities in the function with clear examples. Ideal for Grades 10-12 students studying limits and continuity.

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