Question

Evaluate the given limit. $$ \lim _{x \rightarrow 2} \frac{x^{2}+6 x-16}{x^{2}-3 x+2} $$

Short answer

The limit is 10.

Step-by-step solution

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.

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 \(\frac{(x - 2)(x + 8)}{(x - 2)(x - 1)}\).

Simplify the Expression

After factoring, cancel out the common term \((x - 2)\) from both the numerator and the denominator. The simplified expression is \(\frac{x + 8}{x - 1}\).

Evaluate the Limit of the Simplified Expression

Substitute \(x = 2\) in the simplified expression \(\frac{x + 8}{x - 1}\). This gives \(\frac{2 + 8}{2 - 1} = \frac{10}{1} = 10\).

Key concepts

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’t 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.

Indeterminate forms signal the need for more work to understand the function's behavior near a given point, requiring methods such as factoring to resolve these forms.

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.

By simplifying polynomials through factoring, we can often eliminate components that lead to indeterminate forms, making it easier to evaluate limits.

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.

In the exercise, the expression was simplified from \(\frac{(x - 2)(x + 8)}{(x - 2)(x - 1)}\) to \(\frac{x + 8}{x - 1}\) by canceling \(x - 2\), which was permissible since \(x\) approaches but does not equal 2.

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 \(\frac{10}{1} = 10\).

This straightforward substitution confirms the limit, illustrating the utility of both algebraic manipulation and simplification techniques in limit evaluation.