/*! 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 23 Use algebra to find the limit ex... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 23

Use algebra to find the limit exactly. $$\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}$$

Short Answer

Expert verified
The limit is 4.

Step by step solution

01

Identify the Algebraic Expression

The expression given is \( \frac{x^2 - 4}{x - 2} \). We want to find the limit of this expression as \( x \) approaches 2.
02

Recognize the Indeterminate Form

By substituting \( x = 2 \), the expression becomes \( \frac{4 - 4}{0} \), which is \( \frac{0}{0} \), an indeterminate form. Thus, algebraic manipulation is required.
03

Factor the Numerator

The numerator \( x^2 - 4 \) can be factored as a difference of squares: \( (x - 2)(x + 2) \). So, the expression becomes \( \frac{(x - 2)(x + 2)}{x - 2} \).
04

Simplify the Expression by Canceling Common Terms

Cancel the common factor \( x - 2 \) present in the numerator and the denominator: \( \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \).
05

Evaluate the Limit

Now that the expression is simplified to \( x + 2 \), substitute \( x = 2 \) directly into this replacement expression. Evaluating gives: \( 2 + 2 = 4 \).

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.

Indeterminate Form
In calculus, an **indeterminate form** often signifies a need for further analysis. When evaluating a limit, if substituting the value directly into an expression results in forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), we encounter what is known as an indeterminate form. For the expression \( \frac{x^2 - 4}{x - 2} \), substituting \( x = 2 \) gives \( \frac{0}{0} \), an indeterminate form. This tells us that direct substitution isn't enough, and we must use algebraic techniques to simplify and resolve the expression into something more digestible for evaluating the limit.

  • Indeterminate forms indicate that the limit requires more than straightforward evaluation.
  • They commonly show up in situations where there is division or subtraction of similar terms, leading both the numerator and the denominator to zero.
Algebraic Manipulation
To resolve indeterminate forms, **algebraic manipulation** is key. This process involves restructuring or re-expressing an equation or expression through various algebraic methods until a more straightforward form is achieved. In our original exercise, the expression \( \frac{x^2 - 4}{x - 2} \) needs to be algebraically manipulated since direct evaluation results in an indeterminate form.

Here, algebraic manipulation means:
  • Recognizing specific algebraic patterns, such as the difference of squares, is crucial.
  • Applying these patterns or other algebraic identities to simplify expressions.
  • Canceling out common terms across the numerator and the denominator when possible to simplify the expression further into a manageable form.
After manipulation, you can then substitute the problematic value into the newly simplified expression to find the limit efficiently.
Factoring Polynomials
**Factoring polynomials** is one of the most effective techniques for simplifying expressions in limit calculations. In the given exercise, the polynomial \( x^2 - 4 \) in the numerator can be factored using the difference of squares method. Recognizing such patterns allows you to break down complex expressions into simpler forms.

Let's explore:
  • The difference of squares formula is \( a^2 - b^2 = (a - b)(a + b) \).
  • Using this, \( x^2 - 4 \) can be rewritten as \( (x - 2)(x + 2) \).
  • Factoring transforms the original expression into \( \frac{(x-2)(x+2)}{x-2} \).
  • By canceling the common term \( x - 2 \), the expression simplifies to \( x + 2 \).
Applying these steps not only simplifies the expression but also avoids indeterminate forms, allowing for a straightforward evaluation of the limit once the troublesome term is canceled.

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

Give an explanation for your answer. If \(a\) and \(b\) are positive constants, then \(y=\ln (a x+b)\) has no vertical asymptote.

In \(2014,\) the populations of China and India were approximately 1.355 and 1.255 billion people, \(^{45}\) respectively. However, due to central control the annual population growth rate of China was \(0.44 \%\) while the population of India was growing by \(1.25 \%\) each year. If these growth rates remain constant, when will the population of India exceed that of China?

If \(h(x)=\ln (x+a),\) where \(a>0,\) what is the effect of increasing \(a\) on the vertical asymptote?

Are the statements true or false? Give an explanation for your answer. The function \(g(\theta)=e^{\sin \theta}\) is periodic.

Are the statements true or false? Give an explanation for your answer. $$\sin |x|=|\sin x| \text { for }-2 \pi
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Applied 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.