Problem 89 - Essential Calculus

Question

Average Cost A business has a cost of \(C=0.5 x+500\) for producing \(x\) units. The average cost per unit is \(\bar{C}=\frac{C}{x},\) Find the limit of \(\bar{C}\) as \(x\) approaches infinity.

Step-by-Step Solution

Answer

The limit of \(\bar{C}\) as \(x\) approaches infinity is 0.5

Step 1: Substitute \(C\) in terms of \(x\) into \(\bar{C}\)

In order to find the limit, the first step involves substituting the cost function in terms of \(x\) into the average cost function. Hence, \(\bar{C}=\frac{0.5x + 500}{x}\).

Step 2: Simplify the fraction

The fraction \(\frac{0.5x + 500}{x}\) can be split into two different fractions as \(\frac{0.5x}{x} + \frac{500}{x}\). This simplifies to \(0.5 + \frac{500}{x}\).

Step 3: Find the limit as \(x\) approaches infinity

By the properties of limits, when applying the limit as \(x\) approaches infinity to each term separately, we get \(\lim_{x \to \infty}(0.5) + \lim_{x \to \infty}(\frac{500}{x})\). The first term is simply \(0.5\) since there is no \(x\) present, and the second term goes to \(0\) as any constant divided by infinity results in zero.

Step 4: Compute the final result

From the above calculation, the limit of \(\bar{C}\) as \(x\) approaches infinity is \(0.5 + 0 = 0.5\). So the average cost per unit approaches $0.5 as the number of units produced increases indefinitely.

Key Concepts

Average Cost

Calculating the average cost is a fundamental concept in economics and business. It provides insight into how the expenses associated with production relate to each unit produced. The formula given for the average cost \( \bar{C} \) is \( \bar{C} = \frac{C}{x} \), where \( C \) represents the total cost and \( x \) the number of units. This function helps businesses understand their expenditure efficiency.

In our exercise, the total cost \( C \) is expressed as \( 0.5x + 500 \), indicating a linear relationship. As more units are produced, the cost per unit usually decreases. This is due to spreading fixed costs (like the 500 here, which is likely a fixed cost) over a larger number of units, thereby lowering the average cost.

Understanding average cost isn't just about knowing the formula - it's about understanding how cost dynamics work in production scenarios. Companies seek to lower their average costs through economies of scale, which is highlighted in our task by finding the limit of the average cost as production levels become very large.

Limit of a Function

Limits are a fundamental concept in calculus and essential for understanding how functions behave at extreme values, especially infinity. When we consider limits, we explore what value a function approaches as the input gets exceedingly large or small. In our problem, we need to find the limit of the average cost \( \bar{C} = \frac{0.5x + 500}{x} \) as \( x \to \infty \).

By breaking down the expression, it is rewritten as \( 0.5 + \frac{500}{x} \). The limit of \( 0.5 \) is simply \( 0.5 \), as it does not depend on \( x \) and remains constant. The second term \( \frac{500}{x} \) approaches zero because any non-zero number divided by an increasingly large \( x \) tends towards zero.

This process of evaluating the limit helps verify trends such as diminishing cost impacts in high production scenarios, providing a clearer picture of economic scalability.

Properties of Limits

The properties of limits are crucial for simplifying and calculating limits, like in our problem. One key property used here is the ability to separate and evaluate limits of individual terms within an expression. For any expression that involves addition or subtraction, you can apply the limit to each part separately.

In our example, when we calculated \( \lim_{x \to \infty} (0.5) + \lim_{x \to \infty} \left( \frac{500}{x} \right) \), the result demonstrated this property in action. Knowing these properties allows one to simplify complex functions and discern their behavior as \( x \) approaches certain values.

With these skills, solving limits becomes more manageable, and students can tackle real-world problems, such as optimizing production costs, with greater ease. These properties are not just theoretical—they help us understand practical economic phenomena such as price stabilization and market saturation at very large scales.