/*! 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 47 Prove the limit statements in Ex... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 47

Prove the limit statements in Exercises 37-50. $$\lim _{x \rightarrow 1} f(x)=2 \quad \text { if } \quad f(x)=\left\\{\begin{array}{ll}{4-2 x,} & {x<1} \\ {6 x-4,} & {x \geq 1}\end{array}\right.$$

Short Answer

Expert verified
The limit is \( \lim_{x \to 1} f(x) = 2 \).

Step by step solution

01

Examine the Function and Its Definition

For the given piecewise function \( f(x) \), we have two parts: - \( f(x) = 4 - 2x \) when \( x < 1\) - \( f(x) = 6x - 4 \) when \( x \geq 1\).To prove the limit, we need to ensure that as \( x \) approaches 1 from both sides, the function approaches the same value, which is 2.
02

Calculate the Left-Sided Limit

For \( x < 1 \), substitute into the first part of the function: \( f(x) = 4 - 2x \).We need to find \( \lim_{x \to 1^-} f(x) \):\[\lim_{x \to 1^-} (4 - 2x) = 4 - 2(1) = 2\] Thus, the left-sided limit as \( x \) approaches 1 is 2.
03

Calculate the Right-Sided Limit

For \( x \geq 1 \), use the second part of the function: \( f(x) = 6x - 4 \).The right-sided limit is:\[\lim_{x \to 1^+} (6x - 4) = 6(1) - 4 = 2\]The right-sided limit as \( x \) approaches 1 is also 2.
04

Conclude the Limit Statement

Since the left-sided limit (2) and right-sided limit (2) are equal, we can conclude:\[\lim_{x \to 1} f(x) = 2\]Both sides approaching 2 confirms that the limit exists and equals 2.

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.

Understanding Piecewise Functions
A piecewise function is a function composed of multiple sub-functions, each of which applies to a specific interval of the domain. These functions are represented in smaller segments rather than a single equation throughout. This type of function is useful when different rules need to be applied for different parts of the input. For example, the provided function has two parts:
  • For all values of \( x < 1 \), the function follows the rule \( f(x) = 4 - 2x \).
  • For values of \( x \geq 1 \), the function changes to \( f(x) = 6x - 4 \).
Piecewise functions are common in real-world scenarios where conditions change abruptly, and they allow us to model such situations mathematically. They are crucial for understanding how functions behave in different segments of the input.
Exploring Left-Sided Limits
The left-sided limit of a function at a specific point focuses on approaching the point from the left side on the number line. To find this limit, we consider the values of \( x \) that are slightly less than the point of interest. In this exercise, we calculate \( \lim_{x \to 1^-} f(x) \) using the sub-function that applies when \( x < 1 \), which is \( f(x) = 4 - 2x \). This gives:\[\lim_{x \to 1^-} (4 - 2x) = 4 - 2(1) = 2\]Thus, as \( x \) approaches 1 from the left, the function value gets closer to 2. This calculation helps us understand the behavior of the function immediately before reaching the point \( x = 1 \).
Understanding Right-Sided Limits
The right-sided limit examines the values of \( x \) as they approach a point from the right side. In this case, we look at \( x \geq 1 \), which means considering \( x \) values slightly more than the point in question. For the given problem, the relevant part of the function is \( f(x) = 6x - 4 \). Therefore, we calculate:\[\lim_{x \to 1^+} (6x - 4) = 6(1) - 4 = 2\]This shows that as \( x \) approaches 1 from the right, the function's value also approaches 2. Calculating this limit informs us about the behavior of the function just after crossing the point \( x = 1 \).
Exploring Continuity
Continuity at a point in a function occurs when the following three conditions are met:
  • The function is defined at the point.
  • The left-sided limit and right-sided limit at that point are the same.
  • The common limit value equals the function value at the point.
In this exercise, we found that both the left-sided limit \( (2) \) and the right-sided limit \( (2) \) exist and are equal at \( x = 1 \). Furthermore, since both approaches lead to the same value, the function can be said to be continuous at that point. This means there is no sudden jump or break at \( x = 1 \), providing a smooth transition through this point, which is an essential aspect of understanding the overall behavior of piecewise functions.

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

Find the limits in Exercises \(59-62\) \begin{equation} \lim \left(\frac{1}{t^{3 / 5}}+7\right) \mathrm{as} \end{equation} \begin{equation} \text { a. }t \rightarrow 0^{+} \quad \text { b. } t \rightarrow 0^{-} \end{equation}

Find the limits in Exercises \(49-52\) $$ \lim _{\theta \rightarrow 0}(2-\cot \theta) $$

Suppose that \(f\) is an even function of \(x .\) Does knowing that \(\lim _{x \rightarrow 2^{-}} f(x)=7\) tell you anything about either \(\lim _{x \rightarrow-2^{-}} f(x)\) or \(\lim _{x \rightarrow-2^{+}} f(x) ?\) Give reasons for your answer.

Use the Intermediate Value Theorem to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$ x(x-1)^{2}=1 \quad \text {(one root)} $$

Determine the domain and range of each function. Use various limits to find the asymptotes and the ranges. $$ y=\frac{2 x}{x^{2}-1} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

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.