/*! 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 24 Algebraically determine the limi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 24

Algebraically determine the limits. $$ \lim _{x \rightarrow 2}(3 x+7) $$

Short Answer

Expert verified
The limit is 13.

Step by step solution

01

Understand the Problem

We need to find the limit of the function \(3x + 7\) as \(x\) approaches 2. The limit will tell us what value \(3x + 7\) gets closer to as \(x\) gets closer to 2.
02

Substitute the Value

In this linear function, it is typically valid to directly substitute the \(x\) value into the expression. So, substitute \(x = 2\) into \(3x + 7\).
03

Perform the Calculation

Substitute \(x = 2\) into \(3x + 7\): \[3(2) + 7 = 6 + 7 = 13.\]
04

Conclude the Solution

Since we were able to substitute \(x = 2\) directly, the limit of the function \(3x + 7\) as \(x\) approaches 2 is 13.

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.

Direct substitution method
The direct substitution method is a straightforward technique for evaluating limits. It involves directly substituting the value that \(x\) approaches into the given function. This method is often used when dealing with polynomials or simple functions, such as linear ones. If the substitution does not result in an undefined expression, such as division by zero, this method is often the fastest and simplest way to find a limit.
  • Understand that you can only use this method when the function is continuous at the point \(x\) approaches.
  • In this exercise, substituting 2 into the function \(3x + 7\) was straightforward as it resulted in a defined value without any indeterminacies.
  • Remember, direct substitution will not work for functions that are not continuous at the point in question. For such cases, other limit evaluation techniques must be applied.
By using direct substitution here, we calculated that the limit is 13. This approach is efficient for this type of function.
Linear functions
Linear functions, like \(3x + 7\), are the simplest and most fundamental types of functions in mathematics. They can be recognized by their general form \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. These functions graph as straight lines and have consistent trends.
  • For the function \(3x + 7\), the slope \(m\) is 3, which indicates that for every unit increase in \(x\), \(y\) increases by 3 units.
  • The y-intercept \(b\) is 7, meaning that the line crosses the y-axis at the point (0, 7).
  • Linear functions are continuous everywhere, making limit evaluation straightforward, often using the direct substitution method.
Linear functions are predictable and continuous, making them ideal for practicing basic limits.
Evaluating limits
Evaluating limits is a fundamental concept in calculus. It allows us to understand the behavior of functions as they approach specific points. This concept is crucial for understanding continuity, derivatives, and integrals.
  • When evaluating limits, the first step is to understand the function's behavior near the point of interest鈥攊n this case, \(x = 2\).
  • If the function is continuous at that point, as it is for \(3x + 7\), the limit is simply the function's value at that point.
  • For more complex functions, alternative methods such as factoring, rationalizing, or using L'Hopital's rule may be necessary if direct substitution results in an indeterminate form.
By mastering limit evaluation, students will be able to analyze and interpret the behavior of various functions at specific points.

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

Input and output notation is given for two functions. Determine whether the pair of functions can be combined by function composition. If so, then a. draw an input/output diagram for the new function. b. write a statement for the new function complete with function notation and input and output units and descriptions. The revenue from the sale of \(x\) soccer uniforms is \(R(x)\) yen. The dollar value of \(r\) yen is \(D(r)\) dollars.

Cesium Chloride Cesium chloride is a radioactive substance that is sometimes used in cancer treatments. Cesium chloride has a biological half-life of 4 months. a. Find a model for the amount of cesium chloride left in the body after \(800 \mathrm{mg}\) has been injected. b. How long will it take for the level of cesium chloride to fall below \(5 \% ?\)

The total cost for producing 1000 units of a commodity is \(\$ 4.2\) million, and the revenue generated by the sale of 1000 units is \(\$ 5.3\) million. a. What is the profit on 1000 units of the commodity? b. Assuming \(C(q)\) represents total cost and \(R(q)\) represents revenue for the production and sale of \(q\) units of a commodity, write an expression for profit.

Birth Rate In 1960 , the birth rate for women aged \(15-19\) was 59.9 births per thousand women. In \(2006,\) the birth rate for women aged \(15-19\) was 41.9 births per thousand women. (Source: U.S. National Center for Health Statistics 2006 ) a. Calculate the rate of change in the birth rate for women aged \(15-19\), assuming that the birth rate decreased at a constant rate. b. Write a model for the birth rate for women aged \(15-19\). c. Estimate the birth rate for women aged \(15-19\) in \(2012 .\)

Write a. the sum of the two functions. b. the difference of the first function minus the second function. c. the product of the two functions. d. the quotient of the first function divided by the second function. Evaluate each of these constructed functions at 2. $$ f(x)=5 x+4 ; g(x)=2 x^{2}+7 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.