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.

Short answer

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

Step-by-step solution

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, \(\bar{C}=\frac{0.5x + 500}{x}\).

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 \(0.5 + \frac{500}{x}\).

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.

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.