/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Use a table of values to estimat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 27

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to conirm your result graphically. $$ \lim _{x \rightarrow 0^{+}} x^{x} $$

Short Answer

Expert verified
The limit is 1.

Step by step solution

01

Understand the problem

We need to estimate \( \lim _{x \rightarrow 0^{+}} x^{x} \) using a table of values. This means that we will approach \( x = 0 \) from the positive side and observe the behavior of the function \( x^x \). We will then use these observations to estimate the limit.
02

Create a table of values

Calculate \( x^x \) for values of \( x \) that get progressively closer to 0 from the positive side. For example, evaluate \( x^x \) for \( x = 0.1, 0.01, 0.001, 0.0001, \) and observe the trend of the results.| \( x \) | \( x^x \) ||-------|---------|| 0.1 | 0.794 || 0.01 | 0.954 || 0.001 | 0.993 || 0.0001| 0.999 |
03

Observe the trend

From the table of values, observe that as \( x \) gets closer to 0 from the positive side, the value of \( x^x \) is increasing and getting closer to 1. This suggests that the function approaches the value of 1 as \( x \rightarrow 0^+ \).
04

Graphical confirmation

If possible, use a graphing tool to plot \( x^x \) and look at its behavior as \( x \) approaches 0 from the right. You should notice that the graph is asymptotically approaching a horizontal line at \( y = 1 \) as \( x \rightarrow 0^{+} \).
05

Estimate the limit

From both the table of values and graphic observations, we estimate that the limit is 1. Thus, \( \lim _{x \rightarrow 0^{+}} x^{x} = 1 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Estimating limits
Estimating limits is a technique often used to predict the behavior of a function as it approaches a particular point. In this case, we are examining the limit of the function \( x^x \) as \( x \) approaches zero from the positive side. This means looking at what happens when \( x \) is a very small positive number. By calculating \( x^x \) for progressively smaller values near zero, we observed a trend in the calculated results.
  • For \( x = 0.1 \), we calculated \( x^x = 0.794 \)
  • For \( x = 0.01 \), \( x^x = 0.954 \)
  • For \( x = 0.001 \), \( x^x = 0.993 \)
  • For \( x = 0.0001 \), \( x^x = 0.999 \)
As we see from this table, \( x^x \) values increase and get closer to 1 as \( x \) approaches zero. This systematic way of evaluating values in a table is part of estimating limits and helps make a robust conjecture about the behavior of the function at a point.
Behavior of functions
Understanding the behavior of a function is crucial when estimating limits. In our example of \( x^x \), as \( x \to 0^+ \), observing how the function behaves can give insights into its trend toward an estimated limit.
### How does \( x^x \) behave near zero?When we plot or calculate \( x^x \) for values close to zero, we see it stabilizing near 1. This suggests that the point \( x = 0 \) acts in a specific manner called asymptotic. As \( x \) becomes very small, \( x^x \) appears to hover closer to 1.
  • As \( x \) gets smaller, \( x^x \) rises. This behavior implies stability in the function's output, settling toward a specific limit value, which is 1.
Understanding this behavior provides confidence in predicting limits for functions that exhibit similar patterns.
Graphical analysis
Graphical analysis is a powerful visual tool to verify the estimated behavior of functions, like \( x^x \), as \( x \to 0^+ \). By plotting \( x^x \) on a graphing calculator or software, we can visually confirm our predictions.
### What does the graph show?Upon plotting \( x^x \), you'll notice a curve that gently ascends toward a horizontal line, approaching \( y = 1 \). This visual pattern reflects the numerical progression observed in the table values. The graph essentially shows asymptotic behavior, converging towards 1 as \( x \) nears zero from the right.
  • Identifying this "flattening" at \( y = 1 \) confirms our limit estimate.
  • Graphs offer a tangible way to prove limits observed numerically.
Graphical analysis can bridge a clear understanding, combining visual intuition with calculated estimations, hence making abstract limit concepts more concrete to grasp.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(f(x)=x^{2}\) (a) Estimate the values of \(f^{\prime}(0), f^{\prime}\left(\frac{1}{2}\right), f^{\prime}(1),\) and \(f^{\prime}(2)\) by using a graphing device to zoom in on the graph of \(f .\) (b) Use symmetry to deduce the values of \(f^{\prime}\left(-\frac{1}{2}\right)\) (c) Use the results from parts (a) and (b) to guess a formula for \(f^{\prime}(x)\). (d) Use the definition of derivative to prove that your guess in part (c) is correct.

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \(a\). $$ p(v)=2 \sqrt{3 v^{2}+1}, \quad a=1 $$

Find a formula for a function \(f\) that satisfies the following conditions: $$ \begin{array}{l}{\lim _{x \rightarrow \pm \infty} f(x)=0, \quad \lim _{x \rightarrow 0} f(x)=-\infty, \quad f(2)=0} \\ {\lim _{x \rightarrow 3} f(x)=\infty, \quad \lim _{x \rightarrow 3^{+}} f(x)=-\infty}\end{array} $$

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. $$ f(x)=\left\\{\begin{array}{ll}{x+3} & {\text { if } x \leqslant-1} \\ {2^{x}} & {\text { if } x>-1}\end{array} \quad a=-1\right. $$

Sketch the graph of a function \(f\) that is continuous except for the stated discontinuity. Removable discontinuity at \(3,\) jump discontinuity at 5
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.