Question

Evaluate the given limit. $$ \lim _{x \rightarrow-1} \frac{x^{2}+9 x+8}{x^{2}-6 x-7} $$

Short answer

The limit is \( -\frac{7}{8} \).

Step-by-step solution

Identify the Limit Problem

We need to evaluate the limit \( \lim_{x \rightarrow -1} \frac{x^2 + 9x + 8}{x^2 - 6x - 7} \). This requires simplification of the rational expression before substituting the limit value.

Factorize the Numerator and Denominator

Factor the numerator \( x^2 + 9x + 8 \) and the denominator \( x^2 - 6x - 7 \). The factored form of the numerator is \( (x+1)(x+8) \) and the denominator is \( (x-7)(x+1) \).

Simplify the Rational Expression

Cancel the common factor \( (x+1) \) from the numerator and denominator. The simplified expression is \( \frac{x+8}{x-7} \).

Substitute the Limit Value

Substitute \( x = -1 \) into the simplified expression \( \frac{x+8}{x-7} \). This gives \( \frac{-1+8}{-1-7} = \frac{7}{-8} \).

Conclusion of the Limit Evaluation

The limit \( \lim _{x \rightarrow-1} \frac{x^{2}+9 x+8}{x^{2}-6 x-7} = \frac{7}{-8} \).

Key concepts

Factorization

Factorization is often a crucial step in solving limit problems involving rational expressions. It involves breaking down complex expressions into simpler, more manageable factors. For the given exercise, the numerator is the quadratic expression \( x^2 + 9x + 8 \). By factoring, this becomes \((x+1)(x+8)\). Similarly, the denominator \( x^2 - 6x - 7 \) can be factored into \((x-7)(x+1)\).

The goal of factorization in this context is to reveal any common factors in the numerator and denominator that can be canceled out. This helps in simplifying the expression, making it easier to evaluate the limit. Therefore, recognizing common factors quickly is a key skill in managing these types of problems efficiently.

• Break down complex expressions.
• Simplify the evaluation process.
• Highlight common factors for cancellation.

Rational Expressions

Rational expressions are essentially fractions where both the numerator and the denominator are polynomials. Evaluating the limit of a rational expression, such as \( \frac{x^2 + 9x + 8}{x^2 - 6x - 7} \), requires careful simplification to avoid undefined points due to zero in the denominator.

These expressions often contain complex terms that can be difficult to evaluate under limit conditions if not simplified. By first factorizing both the numerator and the denominator, as shown in the exercise, we can simplify these expressions significantly. This reduces error and simplifies substitution in the next steps.

• Recognize forms of rational expressions.
• Understand the importance of simplification.
• Avoid division by zero issues.

Substitution

Substitution is the process of inserting a specific value into an expression to calculate its limit. After simplifying the rational expression, it is much easier to substitute the given limit value effectively.

In the provided exercise, after canceling out the common factor \((x+1)\), the expression simplifies to \( \frac{x+8}{x-7} \). By substituting \( x = -1 \) into this simpler form, we directly compute the limit as \( \frac{7}{-8} \).

The substitution step confirms the limit's value, ensuring that simplification steps have cleared any potential undefined issues. It is crucial because it directly gives the numeric limit value after plugging the given \( x \) value.

• Replace variable with given limit value.
• Confirm accuracy post-simplification.
• Verify evaluation clarity.

Simplification Steps

Simplification in limit problems is more than just factorizing and canceling terms; it's about making the expression manageable. After factorization, the most critical simplification step in the exercise is the cancellation of the \((x+1)\) factor in both the numerator and the denominator.

This reduces the complexity of the expression from \( \frac{(x+1)(x+8)}{(x-7)(x+1)} \) to \( \frac{x+8}{x-7} \). Simplifying in this manner ensures that when \( x = -1 \) is substituted, we don't encounter undefined situations from the original expression.

Every simplification step should aim to reduce potential algebraic errors and enhance the practice of direct substitution techniques in limits. It builds a fundamental understanding of approaching complex algebraic structures calmly and methodically.

• Facilitate direct substitution.
• Avoid undefined expressions.
• Refine problem-solving skills.