/*! 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 33 Reduce the expression and then e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 33

Reduce the expression and then evaluate the limit. $$ \lim _{x \rightarrow-2}\left(\frac{x^{2}}{x+2}-\frac{4}{x+2}\right) $$

Short Answer

Expert verified
The limit is \( -4 \).

Step by step solution

01

Combine the Fractions

First, notice that both terms in the expression have the common denominator \( x+2 \). By combining them, we get a single fraction: \[ \frac{x^2 - 4}{x+2} \] Here, we simply subtract the numerators to combine the fractions.
02

Factor the Numerator

Now, focus on the numerator \( x^2 - 4 \). This is a difference of squares, which can be factored as follows: \[ x^2 - 4 = (x-2)(x+2) \] Thus, the expression becomes: \[ \frac{(x-2)(x+2)}{x+2} \]
03

Simplify the Expression

Observe that \( x+2 \) is present in both the numerator and denominator. We can cancel \( x+2 \) from both, assuming \( x eq -2 \), to achieve the simplified form: \[ x - 2 \]
04

Evaluate the Limit

Now that the expression is simplified, evaluate the limit as \( x \to -2 \): \[ \lim_{x \to -2} (x - 2) \]Substituting \( x = -2 \) gives us: \[ -2 - 2 = -4 \]

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.

Expression Simplification
When dealing with calculus limits, simplifying expressions is often a crucial first step. You make complex problems simpler, easier to manipulate and solve. In our example, we initially have two separate fractions with a common denominator. We combine these to form one expression: \[ \frac{x^2 - 4}{x+2} \]By combining fractions:
  • Ensure both numerators have the same denominator.
  • Subtract or add the numerators depending on the operation involved.
This forms a single, more manageable fraction, helping us to focus on the actual variable relationships within the limit.
Difference of Squares
The concept of a "difference of squares" is pivotal in factoring polynomials. A difference of squares can be always represented by the formula:\[ a^2 - b^2 = (a - b)(a + b) \]In our exercise, the expression \[ x^2 - 4 \] was identified as a difference of squares since it follows the pattern \[ x^2 - 2^2 \].Factoring is simplified:
  • Recognize the numbers or variables squared.
  • Use the formula to get two binomial products: \( (x - 2)(x + 2) \).
This turned our numerator from a polynomial subtraction into products of linear expressions, which simplifies cancellations.
Limit Evaluation
Limits help us understand the behavior of a function as it approaches a particular point. For this problem, after simplifying the expression to \[ x - 2 \],we find the limit as \( x \) approaches -2.Evaluating limits involves:
  • First simplifying the equation whenever possible.
  • Substituting the value of \( x \) directly, provided no division by zero or undefined expressions occur.
Here, substituting \( x = -2 \) into our simplified expression gives us:\[ -2 - 2 = -4 \]This shows us that as \( x \) gets nearer to -2, the value of our function approaches -4. Proper context and simplification make limits intuitive to evaluate and understand.

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

Explain why \(f\) is continuous on the given interval or intervals $$ f(x)=\sqrt{16 x^{4}-x^{2}} ;\left(-\infty,-\frac{1}{4}\right],\left[\frac{1}{4}, \infty\right) $$

Determine the infinite limit. $$ \lim _{z \rightarrow 3} \frac{2}{(z-3)^{2}} $$

Determine whether \(f\) is continuous or discontinuous at \(a\). If \(f\) is discontinuous at \(a\), determine whether \(f\) is continuous from the right at \(a\), continuous from the left at \(a\), or neither. $$ f(x)=\sqrt{1-e^{x}} ; a=0 $$

Solve the inequality for \(x\). $$ \frac{(x-1)(x-3)}{(2 x+1)(2 x-1)} \geq 0 $$

Decide which of the given one-sided or twosided limits exist as numbers, which as \(\infty\), which as \(-\infty\), and which do not exist. Where the limit is a number, evaluate it. $$ \lim _{x \rightarrow 5^{+}} \frac{1}{x \sqrt{x-5}} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.