Assume postage for sending a first-class letter in the United States is $0.47 for the first ounce (up to and including 1 oz) plus $0.21 for each additional ounce (up to and including each additional ounce).


b. Evaluate lim w→2.3 f(w).

Verified step by step guidance

1. Step 1: Understand the function f(w). The function f(w) represents the cost of sending a letter weighing w ounces. For the first ounce, the cost is $0.47. For each additional ounce, the cost is $0.21.
2. Step 2: Break down the weight w = 2.3 ounces. This weight can be considered as 1 ounce plus an additional 1.3 ounces.
3. Step 3: Calculate the cost for the first ounce. The cost for the first ounce is $0.47.
4. Step 4: Determine the cost for the additional 1.3 ounces. Since the cost is $0.21 for each additional ounce, consider how many full additional ounces are in 1.3 ounces.
5. Step 5: Evaluate the limit as w approaches 2.3. Since the function is piecewise and defined in terms of whole ounces, consider the cost for 2 ounces and the behavior as w approaches 2.3 from both sides.

Key Concepts

Limits

In calculus, a limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for evaluating functions at points where they may not be explicitly defined. For example, evaluating lim w→2.3 f(w) involves determining the value that f(w) approaches as w gets closer to 2.3.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In the context of the postage problem, the cost function can be seen as piecewise, where the cost changes based on the weight of the letter. Understanding how to evaluate limits for piecewise functions is essential, as it may require checking the function's definition at the limit point to ensure continuity.

Continuity

Continuity of a function at a point means that the function is defined at that point, and the limit of the function as it approaches that point equals the function's value at that point. For the limit lim w→2.3 f(w) to exist, f(w) must be continuous at w = 2.3. If the function has a jump or is undefined at that point, the limit may not yield a meaningful result.