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Magazine Chapter 1: Problem 31
Evaluate the given limit. $$ \lim _{x \rightarrow 2} \frac{x^{2}+6 x-16}{x^{2}-3 x+2} $$
Short Answer
Expert verified
The limit is 10.
Step by step solution
01
Identify the Form of the Limit
First, substitute \(x = 2\) into the expression \(\frac{x^2 + 6x - 16}{x^2 - 3x + 2}\). This results in \(\frac{4 + 12 - 16}{4 - 6 + 2}\), which is \(\frac{0}{0}\), an indeterminate form. This signals that further algebraic work is needed.
02
Factor the Numerator and Denominator
The numerator \(x^2 + 6x - 16\) can be factored as \((x - 2)(x + 8)\), and the denominator \(x^2 - 3x + 2\) as \((x - 1)(x - 2)\). Therefore, the expression becomes \(\frac{(x - 2)(x + 8)}{(x - 2)(x - 1)}\).
03
Simplify the Expression
After factoring, cancel out the common term \((x - 2)\) from both the numerator and the denominator. The simplified expression is \(\frac{x + 8}{x - 1}\).
04
Evaluate the Limit of the Simplified Expression
Substitute \(x = 2\) in the simplified expression \(\frac{x + 8}{x - 1}\). This gives \(\frac{2 + 8}{2 - 1} = \frac{10}{1} = 10\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Indeterminate Forms
When working with limits, you might encounter situations where direct substitution of a value into the expression results in forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These are called indeterminate forms. They don鈥檛 provide direct information about the behavior of a function as it approaches a certain point.
- Why Indeterminate? The reason these forms are called indeterminate is because they fail to clearly depict the function's behavior, thus requiring further analysis like algebraic manipulation.
- Next Steps: When faced with an indeterminate form, one approach to finding the limit is to simplify the expression using algebraic techniques such as factoring.
Factoring Polynomials
Factoring is a key algebraic technique in resolving indeterminate forms, especially in polynomial expressions. By changing a polynomial into a product of simpler polynomials, you can often cancel factors, leading to the simplification of a problem.
- Recognizing Patterns: To factor a polynomial effectively, recognize patterns such as the difference of squares or use formulas for sum/product of roots.
- Example Used: In the step-by-step solution, the polynomial \(x^2 + 6x - 16\) factors as \((x - 2)(x + 8)\). This was achieved by identifying numbers that multiply to -16 and add to 6.
Simplifying Rational Expressions
Once polynomials are factored, the next step is simplifying rational expressions. This involves canceling out common factors in the numerator and denominator.
- Identifying Common Factors: Look for terms that appear both as a factor in the numerator and denominator. These can be canceled to reduce the expression.
- Risks in Cancellation: Canceled terms must not equal zero at the point of interest, as this would invalidate the simplification.
Evaluating Limits
The final step in resolving a limit question is to evaluate the limit of the simplified expression. This involves substituting the limit point into the simplified version of the function.
- Direct Substitution: If simplifying the expression removes indeterminate forms, you can substitute the value directly into the simplified expression.
- Conclusion in Example: In our exercise, the simplified expression \(\frac{x + 8}{x - 1}\) was used to find the limit as \(x\) approaches 2, giving \(\frac{10}{1} = 10\).
