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Chapter 3: Problem 26

Use the rules of limits to find the indicated limits if they exist. Support your answer using a computer or graphing calculator. $$ \lim _{x \rightarrow 1} \pi $$

Short Answer

Expert verified
The limit is \( \pi \).

Step by step solution

01

Understand the Problem

We need to find the limit of a constant, namely \( \pi \), as \( x \) approaches 1. In mathematical terms, this is written as: \( \lim_{x \to 1} \pi \).
02

Apply the Limit Properties to a Constant

The limit of a constant does not depend on the variable. The property states that for a constant \( c \), \( \lim_{x \to a} c = c \). This is because a constant doesn鈥檛 change as \( x \) changes鈥攊t remains the same.
03

Solve Using Limit Rule

Since \( \pi \) is a constant, we apply the property: \( \lim_{x \to 1} \pi = \pi \). The limit of \( \pi \) as \( x \) approaches 1 is simply \( \pi \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Functions
Constant functions are quite simple to understand. They are functions that do not change, regardless of the input value. For example, the number \( \pi \) is a constant function. It remains the same no matter what \( x \) value you choose. Constant functions can be represented as \( f(x) = c \), where \( c \) is a constant number.
  • If your function is always the same number, then it's a constant function.
  • No matter where you are on the graph, the function value stays constant.
Understanding constant functions is crucial because they help us develop a fundamental understanding of limits and how they behave. When we explore limits in calculus with constant functions, these can be viewed as flat horizontal lines on a graph. This means the limit of a constant function as \( x \) approaches any number \( a \) is simply the constant itself.
Limit Properties
Limits have specific properties that simplify working with them, especially in calculus. One of these is the **Constant Rule for Limits**, which states:\[\lim_{x \to a} c = c\]This rule tells us that the limit of a constant \( c \) as \( x \) approaches any number \( a \) will always return the constant \( c \). It doesn't matter what \( a \) is, because the constant doesn't change.
  • This property is quite intuitive since constants have no dependency on \( x \).
  • Applying this property makes problems involving constant functions straightforward.

    • Using Limit Properties

  • When working through problems, always consider if a limit property can be applied to simplify your work.
  • This specific property is often the first step in solving limits involving constants.
Graphing Calculators
Graphing calculators are powerful tools used in mathematics to visualize and solve problems involving limits among other computations.When you use a graphing calculator to find the limit of a constant function, you can directly input the function and set the variable \( x \) to approach a particular value. In these cases, the calculator reaffirms what we know: the limit of a constant is the constant itself.
  • Graphing calculators help students see functions and changes in real-time.
  • They can confirm results derived analytically and make complex equations more comprehensible.
By visually examining the graph of a constant function, you're observing a horizontal line. As \( x \) moves towards any number, the graph provides a visual confirmation that the function's value remains the same.

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Most popular questions from this chapter

Let \(f(x)=\frac{|x|}{x}\) if \(x \neq 0 .\) Is it possible to define \(f(0)\) so that \(\lim _{x \rightarrow 0} f(x)\) exists? Why or why not?

Cost Curve Dean \(^{69}\) made a statistical estimation of the cost-output relationship for a hosiery mill. The data for the firm is given in the following table. $$ \begin{array}{|l|llllll|} \hline x & 45 & 56 & 62 & 70 & 74 & 78 \\ \hline y & 14 & 17 & 19 & 20.5 & 21.5 & 22.5 \\ \hline \end{array} $$ Here \(x\) is production in hundreds of dozens, and \(y\) is the total cost is thousands of dollars. a. Determine both the best-fitting quadratic using least squares and the square of the correlation coefficient. Graph. b. Using the quadratic cost function found in part (a) and the approximate derivative found on your computer or graphing calculator, graph the marginal cost. What is happening to marginal cost as output increases? Explain what this means.

Learning The time \(T\) it takes to memorize a list of \(n\) items is given by \(T(n)=3 n \sqrt{n-3}\). Find the approximate change in time required from memorizing a list of 12 items to memorizing a list of 14 items. Find the slope of the tangent line at \(n=12\) by using your computer or graphing calculator to plot the tangent line at the appropriate point, or use a screen with smaller and smaller dimensions until the graph looks like a straight line. (See the Technology Resource Manual.)

Biology An anatomist wishes to measure the surface area of a bone that is assumed to be a cylinder. The length \(l\) can be measured with great accuracy and can be assumed to be known exactly. However, the radius of the bone varies slightly throughout its length, making it difficult to decide what the radius should be. Discuss how slight changes in the radius \(r\) will approximately affect the surface area.

Economic Growth and the Environment Grossman and Krueger \(^{41}\) studied the relationship in a variety of countries between per capita income and various environmental indicators. The object was to determine whether environmental quality deteriorates steadily with growth. They found that the equation $$ y=f(x)=0.27 x^{3}-6.31 x^{2}+39.87 x+34.78 $$ approximated the relationship between \(x\) given as GDP per capita income in thousands of dollars and \(y\) given as units of smoke in cities. Graph on your computer or graphing calculator using a screen with dimensions [0,9.4] by \([0,150] .\) Have your computer or graphing calculator draw tangent lines to the curve when \(x\) is \(2,3,5,\) and \(6 .\) Note the slope, and relate this to the rate of change. Interpret what each of these numbers means. On the basis of this model. determine whether environmental quality deteriorates with economic growth.
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