Math / Calculus

QuestionFirst-class postage is $\$ 0.34$ for the first ounce (or any fraction thereof) and $\$ 0.25$ for each additional ounce (or fraction thereof). Let $C(x)$ represent the postage for a letter weighing $\times$ oz. Use this information to answer the questions. a) Find $\lim C(x)$ . Select the correct choice below and fill in any answer boxes in your choice. $x \rightarrow 3^{-}$ A. $\lim _{x \rightarrow 3^{-}} C(x)=\$ \square$ (Type an integer or a decimal.) $\square$ B. The limit does not exist.

Studdy Solution

STEP 1

1. The cost function $C(x)$ is defined piecewise based on the weight $x$ of the letter in ounces.
2. The cost is \ $0.34 for the first ounce or any fraction thereof.<br />3. For each additional ounce or fraction thereof, the cost is an additional \$0.25 per ounce.<br />4. We need to evaluate the limit$ \lim_{x \rightarrow 3^{-}} C(x) $, where$ x$ is approaching 3 from the left.

STEP 2

1. Define the cost function $C(x)$ based on the given pricing structure.
2. Determine the cost for weights approaching 3 ounces from the left.
3. Evaluate the limit $\lim_{x \rightarrow 3^{-}} C(x)$ .

STEP 3

Define the cost function $C(x)$ based on the given pricing structure.
$C(x) = \begin{cases} \$0.34 & \text{for } 0 < x \leq 1 \\ \$0.34 + \$0.25 & \text{for } 1 < x \leq 2 \\ \$0.34 + 2 \times \$0.25 & \text{for } 2 < x \leq 3 \\ \end{cases}$

STEP 4

Determine the cost for weights approaching 3 ounces from the left. This means we consider the interval $2 < x < 3$ .
For $2 < x < 3$ : $C(x) = \$0.34 + 2 \times \$0.25 = \$0.34 + \$0.50 = \$0.84$

STEP 5

Evaluate the limit $\lim_{x \rightarrow 3^{-}} C(x)$ .
Since $C(x)$ is constant in the interval $2 < x < 3$ , as $x$ approaches 3 from the left, $C(x)$ remains $\$0.84$ .
$\lim_{x \rightarrow 3^{-}} C(x) = 0.84$
Therefore, the answer to the problem is:
$\boxed{\lim_{x \rightarrow 3^{-}} C(x) = \$0.84}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms