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Chapter 2: Problem 21

find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{u \rightarrow-2} \frac{u^{2}-u x+2 u-2 x}{u^{2}-u-6} $$

Short Answer

Expert verified
The limit is \( \frac{-2-x}{-5} \).

Step by step solution

01

Analyze the Expression

Examine the given limit \( \lim _{u \rightarrow -2} \frac{u^{2} - u x + 2 u - 2 x}{u^{2} - u - 6} \). We notice that directly substituting \( u = -2 \) into the expression will result in a \( \frac{0}{0} \) indeterminate form, so we must perform algebraic manipulation to simplify the expression.
02

Factor the Denominator

The denominator is \( u^{2} - u - 6 \). We factor it to find the roots: \[(u - 3)(u + 2)\]This cancels out at the point \( u = -2 \), suggesting there might be a common factor in the numerator.
03

Simplify the Numerator

Rewrite the numerator to identify common factors: \( u^{2} - u x + 2 u - 2 x \). Group the terms: \( (u^{2} + 2u) - (ux + 2x) \). Factor both groups:\[u(u + 2) - x(u + 2) = (u + 2)(u - x)\]
04

Cancel Common Factors

Now the expression becomes: \[ \lim_{u \rightarrow -2} \frac{(u + 2)(u - x)}{(u - 3)(u + 2)} \]. Cancel the common factor \((u + 2)\) from the numerator and the denominator.
05

Evaluate the Simplified Limit

Substitute \( u = -2 \) in the remaining expression: \[ \lim_{u \rightarrow -2} \frac{u - x}{u - 3} = \frac{-2 - x}{-2 - 3} = \frac{-2 - x}{-5} \]. This result is simplified.

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation is a crucial technique in calculus, especially when dealing with limit problems. In many cases, substitution might first lead to an indeterminate form such as \( \frac{0}{0} \). This requires us to manipulate the expression algebraically to simplify it and avoid problematic forms.
  • Start by rearranging terms or grouping them to make them easier to factor.
  • Identify common patterns or factors that can cancel out to simplify complex expressions.
By performing these initial steps, you can transform the expression into a form that makes it possible to evaluate the limit correctly.
Indeterminate Forms
Indeterminate forms, such as \( \frac{0}{0} \), occur in calculus when substitution into a limit results in an undefined expression. These forms signal that the limit needs further evaluation since it cannot be directly calculated from the original equation.
  • These forms raise the need for additional algebraic manipulation or other methods to find a defined result.
  • They often require recognizing special patterns or applying calculus techniques to resolve the expression into a determinate form.
Understanding indeterminate forms is key to effectively using limit evaluation techniques because it guides how to approach problem-solving in calculus.
Factoring Polynomials
Factoring polynomials is essential when simplifying expressions in calculus, especially for limits. It is the process of rewriting a polynomial as a product of its factors, making simplification easier.
  • This technique is beneficial for identifying common factors in both the numerator and denominator of an expression.
  • By successfully factoring, you can often cancel these common factors, which significantly simplifies the calculations involved.
  • Recognizing standard factoring patterns, like difference of squares or trinomial factoring, helps in performing these actions quickly and accurately.
In limits, factoring polynomials is a go-to tool for handling expressions that initially yield indeterminate forms.
Limit Evaluation Techniques
Limit evaluation techniques are methods used to find the limit of an expression as the variable approaches a certain value. These techniques become necessary when direct substitution is not suitable due to indeterminate forms.
  • Factoring: Simplifies the expression by eliminating common factors, as shown in the given exercise.
  • Substitution: After simplification, direct substitution often provides the limit.
  • L'H么pital's Rule: Used for \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) forms by differentiating the numerator and denominator separately.
  • Simplification Techniques: May involve expanding, combining fractions, or performing algebraic manipulations to reach a solvable form.
Employing these techniques correctly allows you to solve limits effectively, ensuring that all expressions can be simplified to a form where the limit exists or can be clearly identified.

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