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Chapter 2: Problem 74

Find each limit, if it exists. $$\lim _{x \rightarrow-2} \frac{x^{3}+8}{x^{2}-4}$$

Short Answer

Expert verified
The limit is \( -3 \)

Step by step solution

01

Simplify the Expression

First, simplify the given expression. The numerator can be factored using the sum of cubes formula: \[ x^3 + 8 = (x + 2)(x^2 - 2x + 4) \] and the denominator can be factored as a difference of squares: \[ x^2 - 4 = (x - 2)(x + 2) \]
02

Write the Factored Form

Rewrite the limit using the factored forms: \[\frac{(x + 2)(x^2 - 2x + 4)}{(x - 2)(x + 2)}\]
03

Cancel Common Factors

Next, cancel the common factor \( (x + 2) \) in the numerator and denominator: \[\frac{x^2 - 2x + 4}{x - 2}\]
04

Apply the Limit

Finally, apply the limit by substituting \( x \rightarrow -2 \) into the simplified expression: \[ \frac{(-2)^2 - 2(-2) + 4}{-2 - 2} = \frac{4 + 4 + 4}{-4} = \frac{12}{-4} = -3\]

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Key Concepts

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

Factoring Polynomials
Factoring polynomials involves writing a polynomial as a product of its binomials or trinomials. This process simplifies complex expressions and is crucial in solving various algebraic problems. For example, consider the polynomial expression in our limit calculation:
  • The numerator: \(x^3 + 8\)
  • The denominator: \(x^2 - 4\)
Writing these polynomials in their factored forms helps identify common factors that can be canceled out. Factoring is often the first step when dealing with polynomial limits.
Sum of Cubes
The sum of cubes is a special factoring formula used when you have terms like \(x^3 + y^3\). This can be factored as: \ { x^3 + y^3 = (x + y)(x^2 - xy + y^2) } In the given exercise, the numerator \( x^3 + 8 \) is a sum of cubes because \(8\) can be expressed as \(2^3\). Therefore: \ x^3 + 8 = (x + 2)(x^2 - 2x + 4) This formula provides a clear structure, making it easier to manage complex expressions.
Difference of Squares
The difference of squares formula applies when you have an expression like \(a^2 - b^2\). This can be factored as: \ { a^2 - b^2 = (a + b)(a - b) } In the limit problem, the denominator \(x^2 - 4\) is a difference of squares because \(4\) can be written as \(2^2\). Therefore: \ x^2 - 4 = (x + 2)(x - 2) Using this factoring formula helps break down expressions into more manageable parts for solving limits.
Canceling Common Factors
After factoring both the numerator and the denominator, you often find common factors that can be canceled. This is a crucial step in simplifying expressions before applying limits. In our exercise, after factoring we have: \ { \frac{(x + 2)(x^2 - 2x + 4)}{(x - 2)(x + 2)} } The common factor \( (x + 2) \) appears in both the numerator and the denominator, so they cancel each other out, simplifying our limit expression to: \ { \frac{x^2 - 2x + 4}{x - 2} }This step makes it easier to apply the limit \( x \rightarrow -2 \) without any issues of undefined forms.

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Most popular questions from this chapter

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