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Chapter 2: Problem 39

In the following exercises, set up a table of values to find the indicated limit. Round to eight digits. $$ \lim _{x \rightarrow 1}(1-2 x) $$

Short Answer

Expert verified
The limit is -1.

Step by step solution

01

Identify the Function

We need to find the limit of the function \( f(x) = 1 - 2x \) as \( x \) approaches 1. The target value is \( x = 1 \).
02

Choose Values Close to 1

To approximate the limit, we choose values of \( x \) that are close to 1 both from the left (e.g., 0.9, 0.99, 0.999) and from the right (e.g., 1.1, 1.01, 1.001).
03

Calculate the Function Values

Compute the value of \( 1-2x \) for each chosen \( x \): - For \( x = 0.9 \), \( f(x) = 1 - 2(0.9) = 1 - 1.8 = -0.8 \) - For \( x = 0.99 \), \( f(x) = 1 - 2(0.99) = 1 - 1.98 = -0.98 \) - For \( x = 0.999 \), \( f(x) = 1 - 2(0.999) = 1 - 1.998 = -0.998 \) - For \( x = 1.1 \), \( f(x) = 1 - 2(1.1) = 1 - 2.2 = -1.2 \) - For \( x = 1.01 \), \( f(x) = 1 - 2(1.01) = 1 - 2.02 = -1.02 \) - For \( x = 1.001 \), \( f(x) = 1 - 2(1.001) = 1 - 2.002 = -1.002 \)
04

Analyze the Table of Values

Create a table with the \( x \) and corresponding \( f(x) \) values:\[\begin{array}{|c|c|}\hlinex & f(x) \\hline0.9 & -0.8 \0.99 & -0.98 \0.999 & -0.998 \1.001 & -1.002 \1.01 & -1.02 \1.1 & -1.2 \\hline\end{array}\] Notice that as \( x \) approaches 1, \( f(x) \) tends toward -1.
05

Conclude the Limit

Based on the table of values, as \( x \) approaches 1, the value of \( f(x) \) approaches -1. Thus, \(\lim _{x \rightarrow 1}(1 - 2x) = -1.\)

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Key Concepts

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

Function Evaluation
In calculus, function evaluation is the process of determining the output of a function for a specified input. For the exercise given, you are dealing with the function \( f(x) = 1 - 2x \). Evaluating this function means calculating \( f(x) \) for different values of \( x \). This involves substituting the value of \( x \) into the equation, performing the arithmetic operation. For example, when \( x = 0.9 \), substituting gives \( f(0.9) = 1 - 2(0.9) = -0.8 \). Function evaluation helps to understand how changes in \( x \) affect \( f(x) \). It's the foundation of limit finding, as you explore the behavior of \( f(x) \) near a specific point.
Approaching Values
In limits, 'approaching values' refers to the process of getting closer and closer to a particular point on the \( x \)-axis but never actually reaching it. For the limit \( \lim _{x \rightarrow 1}(1 - 2x) \), you are interested in seeing how \( f(x) \) behaves as \( x \) gets very close to 1, without necessarily being exactly 1. Imagine standing at a crossroads and taking tiny steps towards a particular junction, observing changes as you move. As you approach 1 from either direction (either incrementally decreasing from the right or increasing from the left), you analyze how \( f(x) \) behaves to determine its limiting value.
Table of Values
A table of values is a simple yet effective tool to visualize how a function behaves as \( x \) approaches a target value. In this exercise, a table is used to show various \( x \) values closely surrounding 1, both from the left (e.g., 0.9, 0.99, 0.999) and from the right (e.g., 1.1, 1.01, 1.001). For each \( x \), you compute \( f(x) \) and note the results:
  • 0.9 gives -0.8
  • 0.99 gives -0.98
  • 0.999 gives -0.998
  • 1.1 gives -1.2
  • 1.01 gives -1.02
  • 1.001 gives -1.002
Analyzing this table, you observe that as \( x \) nears 1, \( f(x) \) draws closer to -1. This confirms the limit prediction of the function's behavior near the specified point.
Limit Calculation Steps
To calculate the limit, follow specific steps to ensure accuracy and completeness.
  • Identify the Function: Recognize the function you're evaluating, such as \( f(x) = 1 - 2x \), for which you find the limit as \( x \) approaches a certain value.
  • Choose Values Around the Target: Select values incrementally nearer to the target value of \( x = 1 \) from both sides (left and right).
  • Evaluate the Function: Determine \( f(x) \) for each chosen \( x \). This helps understand the approaching behavior of the function.
  • Analyze the Results: Look at the trends in \( f(x) \) as \( x \) gets close to the target. Here, you observed \( f(x) \) nears -1 as \( x \) approaches 1.
  • Conclude: Based on observations, formalize the limit statement, such as \( \lim _{x \rightarrow 1}(1 - 2x) = -1 \).
These steps showcase a structured approach in limit evaluation, making it an organized process to derive reliable results.

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Most popular questions from this chapter

In the following exercises, sketch the graph of a function with the given properties. $$ \begin{array}{l}{\lim _{x \rightarrow-\infty} f(x)=0, \lim _{x \rightarrow-1^{-}} f(x)=\infty, \lim _{x \rightarrow-1^{+}} f(x)=-\infty} \\\ {f(0)=-1, \lim _{x \rightarrow 1^{-}} f(x)=-\infty, \lim _{x \rightarrow 1^{+}} f(x)=\infty, \lim _{x \rightarrow \infty} f(x)=0}\end{array} $$

[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits. \(f(x)=\left\\{\begin{array}{ll}{x^{2},} & {x \leq 3} \\ {x+4,} & {x>3}\end{array}\right.\) $$a. \quad \lim _{x \rightarrow 3^{-}} f(x)$$ $$b. \quad \lim _{x \rightarrow 3^{+}} f(x)$$

T] After a certain distance \(D\) has passed, the gravitational effect of Earth becomes quite negligible, so we can approximate the force function by $$ F(d)=\left\\{\begin{array}{ll}{-\frac{m k}{d^{2}}} & {\text { if } d

\([\mathrm{T}]\) In the following exercises, use a graphing calculator to find a number \(\delta\) such that the statements hold true. \(|\sqrt{x-4}-2|<0.1,\) whenever \(|x-8|<\delta\)

If the left- and right-hand limits of \(f(x)\) as \(x \rightarrow a\) exist and are equal, then \(f\) cannot be discontinuous at \(x=a .\)
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