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The cost (in dollars) for manufacturing a particular flashdrive is C(x) = 13,000 + 7x, where x is the number of flashdrives produced. Recall from the previous chapter that the average cost

per flashdrive, denoted by C(x), is found by dividing C(x) by x. Find and interpret lim C(x).

X→00

The problem provides the total cost function $ C(x) = 13,000 + 7x $, where $ x $ represents the number of flash drives produced. We are asked to determine the average cost function $ \bar{C}(x) $ and find the limit as $ x $ approaches infinity.

### Step 1: Determine the Average Cost Function $ \bar{C}(x) $
The average cost per flash drive is given by:
\[
\bar{C}(x) = \frac{C(x)}{x}
\]
Substituting the cost function $ C(x) = 13,000 + 7x $:
\[
\bar{C}(x) = \frac{13,000 + 7x}{x}
\]
This can be simplified as:
\[
\bar{C}(x) = \frac{13,000}{x} + 7
\]

### Step 2: Find and Interpret $ \lim_{x \to \infty} \bar{C}(x) $
We need to compute the limit of the average cost function as $ x $ approaches infinity:
\[
\lim_{x \to \infty} \bar{C}(x) = \lim_{x \to \infty} \left( \frac{13,000}{x} + 7 \right)
\]
As $ x $ becomes very large, the term $ \frac{13,000}{x} $ approaches 0. Therefore, the limit is:
\[
\lim_{x \to \infty} \bar{C}(x) = 7
\]

### Interpretation
As the number of flash drives produced becomes very large, the average cost per flash drive approaches $7. This makes sense because the fixed cost (13,000) becomes negligible as production increases, leaving only the variable cost per flash drive.

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Do you have any further questions or need more details?

Here are 5 related questions you could explore:
1. What would happen to the average cost function if the fixed cost increased?
2. How does the concept of economies of scale relate to the limit of the average cost function?
3. How can you calculate the break-even point for this cost function?
4. What would the graph of the average cost function look like?
5. How does the variable cost per unit affect the long-term average cost?

**Tip:** When analyzing average cost functions, look for how fixed costs influence the behavior of the function at different production levels.

Explore how to find the limit of the average cost function C(x) = (13,000 + 7x) / x as x approaches infinity. Understand the concept of average cost and its behavior in manufacturing economics.

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