Problem 83 - Precalculus with Limits

Question

CONSUMER AWARENESS The cost of hooking up and towing a car is \(\$85\) for the first mile and \(\5\) for each additional mile or portion of a mile. A model for the cost \(C\) (in dollars) is \(C(x)=85 - 5[[-(x-1)]], where \)x\( is the distance in miles. (Recall from Section 1.6 that \)f(x) = [[x]] = \( the greatest integer less than or equal to \)x\(). (a) Use a graphing utility to graph \)C\( for \)0 < x \leq 10\(. (b) Complete the table and observe the behavior of \)C\( as \)x\( approaches 5.5. Use the graph from part (a) and the table to find \)\lim_{x \to 5.5}\ C(x)\(. (c) Complete the table and observe the behavior of \)C\( as \)x\( approaches 5. Does the limit of \)C(x)\( as \)x$ approaches 5 exist? Explain.

Step-by-Step Solution

Answer

The cost function \(C(x)\) jumps steps at every integer value of \(x\). Therefore, for \(x=5.5\), the limit exists and equals the cost for 6 miles, but for \(x=5\), the limit does not exist because the left and right limits do not coincide.

Step 1: Graphing \(C(x)\)

A graphing utility (like a graphing calculator or a mathematical software) is used to plot \(C(x) = 85 + 5[[-(x-1)]]\) for \(0 < x \leq 10\). The graph represents the cost for each mile.

Step 2: Populate the Table and Calculate Limit for \(x=5.5\)

Calculate the cost for each mile, especially those close to 5.5. Then use them to find \(\lim_{x \to 5.5}\) \(C(x)\). Which should give a specific cost that looks like a step increase.

Step 3: Populate the Table and Analyze Limit for \(x=5\)

Compute the cost for each mile around x=5. but This time, the limit does not exist because the cost 'jumps' from one value to another at x=5. That is, \(\lim_{x \to 5^{-}} C(x) \neq \lim_{x \to 5^{+}} C(x)\). And because the left-limit and right-limit do not equal, the limit at x=5 does not exist.

Key Concepts

Understanding Step Functions

Step functions are special types of piecewise functions. They look like a staircase on a graph because they have jumps or "steps."

In our exercise, the cost function \(C(x)\) is modeled as a step function. Here's how it works:

• The initial cost is \(\\(85\) for the first mile.
• Each additional mile or portion is charged \(\\)5\).

This results in a sudden increase or "step-up" in cost for each new mile.

The mathematical representation uses the greatest integer function \([[-(x-1)]]\), which tracks the whole number parts of distances. This essentially groups distances into segments, creating clear jumps in the function.

Understanding step functions helps you predict where these jumps occur and how they affect the overall graph of the function.

Using a Graphing Utility

A graphing utility is a tool like a graphing calculator or software used to plot functions. In the context of this exercise, it is essential for visualizing the piecewise nature of \(C(x)\).

When you use a graphing utility:

• Input the function \(C(x) = 85 + 5[[-(x-1)]]\).
• Set the domain of interest, here from \(0 < x \leq 10\).
• Observe the plotted steps that demonstrate the cost structure.

This visual representation helps in understanding how the cost increases in steps as distance increases. It also shows where the limits are, providing a clear picture of the step function's behavior. Graphing utilities allow you to see numerical trends and verify calculations immediately.

Exploring Limit in Calculus

Limits help us understand what happens to a function as we approach a certain point. In piecewise or step functions, limits can show potential discontinuities.

In our exercise, we explore limits as \(x\) approaches \(5.5\) and \(5\).

For \(x = 5.5\):

• Since this is not at a point of discontinuity, the limit \(\lim_{x \to 5.5} C(x)\) exists, representing the consistent cost right before and after \(x = 5.5\).

However, at \(x = 5\):

• The cost jumps, and the left-hand limit \(\lim_{x \to 5^{-}} C(x)\) is not equal to the right-hand limit \(\lim_{x \to 5^{+}} C(x)\).
• This means the limit as \(x\) approaches \(5\) does not exist.

Recognizing these points of discontinuity is key in understanding the behavior of step functions and calculating limits.