Problem 75 - Calculus and its applications

Question

The cost of sending a large envelope via U.S. first-class mail in 2014 was \(\$ 0.98\) for the first ounce and \(\$ 0.21\) for each additional ounce (or fraction thereof). (Source: www.usps.com.) If \(x\) represents the weight of a large envelope, in ounces, then \(p(x)\) is the cost of mailing it, where $$ \begin{array}{l} p(x)=\$ 0.98, \text { if } \quad 0 < x \leq 1, \\ p(x)=\$ 1.19, \text { if } \quad 1 < x \leq 2, \\ p(x)=\$ 1.40, \text { if } 2 < x \leq 3, \end{array} $$ and so on, up through 13 ounces. The graph of \(p\) is shown below. Using the graph of the postage function, find each of the following limits, if it exists. $$ \lim _{x \rightarrow 3} p(x) $$

Step-by-Step Solution

Answer

The limit \( \lim_{x \rightarrow 3} p(x) \) does not exist because the left and right limits are not equal.

Step 1: Understanding the Problem

We need to find the limit of the postage cost function \( p(x) \) as \( x \) approaches 3 ounces. The cost function \( p(x) \) is defined as a step function that changes value at each whole number of ounces.

Step 2: Identify Limit from the Left

Since \( x \rightarrow 3 \) implies checking the function value just before \( x = 3 \), we examine \( 2 < x \leq 3 \). In this interval, the cost \( p(x) \) is \( \$1.40 \).

Step 3: Identify Limit from the Right

For \( x \rightarrow 3 \) from values slightly greater than 3, we have the interval \( 3 < x \leq 4 \). In this interval, the function value is \( p(x) = \$1.61 \).

Step 4: Consider Left and Right Limits

For a limit to exist as \( x \rightarrow 3 \), the left-hand limit \( \lim_{x \rightarrow 3^-} p(x) \) must equal the right-hand limit \( \lim_{x \rightarrow 3^+} p(x) \).

Step 5: Compare Limits to Determine Existence

The left-hand limit is \( \\(1.40 \) and the right-hand limit is \( \\)1.61 \). Since these two are not equal, the limit \( \lim_{x \rightarrow 3} p(x) \) does not exist.

Key Concepts

Step Function

A step function is a type of function in mathematics where there is a series of distinct, constant outputs for various intervals of the input. This means that as the input value changes within an interval, the output value remains the same. However, as soon as the input value crosses the boundary of an interval, the output value suddenly changes to a different constant value.

For example, if we look at our postage cost function, it's constructed using a step function. Here, each segment pertains to a weight range of the envelope. Within each weight range, no matter how much the weight changes, as long as it stays within the range, the cost remains constant. But, if the weight moves even slightly into the next range, the cost immediately jumps to the next predefined rate.

This concept is commonly used in real-world scenarios, such as billing systems or transport costs, where the rate is fixed for a specific range of use or consumption.

Postage Cost Function

The postage cost function for mailing a large envelope is a practical example of a step function. Postage costs in this situation do not change smoothly with weight. Instead, costs increase suddenly at the upper boundary of each weight category.

In this case,

• The first ounce costs $0.98.
• The second ounce costs $1.19.
• The third ounce costs $1.40.

And this pattern continues for each additional ounce up to 13 ounces. This function exemplifies a step function where the input (weight) being a little more, results in the output (cost) jumping to the next level. It's essential for functions like this to be understood in their stepwise nature to accurately predict costs.

This method helps entities like postal services map out predictable, simple-to-understand cost structures that are easy for users and consumers to follow.

Left-Hand Limit

In calculus, the left-hand limit refers to the value that a function approaches as the input approaches a particular point from the lower side of that point. Essentially, it's as if you're sneaking up to that point from the left.

In our example of the postage cost function, when determining the limit as the envelope weight approaches 3 ounces, we consider what happens as the weight comes from slightly less than 3 ounces. For the interval just under 3 ounces, which is 2 to 3 ounces, the postage cost is $1.40. Hence, the left-hand limit at this point is \(\lim_{x \to 3^-} p(x) = 1.40.\)

This concept is crucial for understanding discontinuous functions like step functions, where values reading from one direction (left) might differ from those read approaching from the other (right).

Right-Hand Limit

The right-hand limit in calculus refers to the value a function approaches as the input approaches a particular point from the higher side of that point. It's as if you're coming at the point from the right.

With the postage cost example, to find the limit as the envelope weight approaches 3 ounces from above, we examine the weight interval just more than 3 ounces. In this case, the range from 3 to 4 ounces costs $1.61. Thus, the right-hand limit at this point is \(\lim_{x \to 3^+} p(x) = 1.61.\)

Recognizing the difference between left and right-hand limits is essential when dealing with piecewise or step functions, especially when the function doesn't match up perfectly at a point, indicating the overall limit at that point does not exist. This understanding is vital for describing the behavior of a function around points of discontinuity.