Thank you for the clarification! Let's correct the solution given that \(x\) is approaching \(-7\).

We are given the limit:

\[
\lim_{{x \to -7}} \frac{x^2 + 7x}{x^2 - 2x - 63}
\]

### Step 1: Factor the denominator
As before, let's factor the denominator \(x^2 - 2x - 63\):

\[
x^2 - 2x - 63 = (x - 9)(x + 7)
\]

### Step 2: Simplify the expression
The numerator \(x^2 + 7x\) can also be factored:

\[
x^2 + 7x = x(x + 7)
\]

So the limit simplifies to:

\[
\lim_{{x \to -7}} \frac{x(x + 7)}{(x - 9)(x + 7)}
\]

We can cancel out the common factor \((x + 7)\) in the numerator and denominator:

\[
\lim_{{x \to -7}} \frac{x}{x - 9}
\]

### Step 3: Substitute \(x = -7\)
Now, substitute \(x = -7\) into the simplified expression:

\[
\frac{-7}{-7 - 9} = \frac{-7}{-16} = \frac{7}{16}
\]

### Final Answer
The limit is:

\[
\lim_{{x \to -7}} \frac{x^2 + 7x}{x^2 - 2x - 63} = \frac{7}{16}
\]

Would you like more details or have any other questions?

Here are some related questions you might find helpful:
1. What happens when you cancel common factors in limits?
2. How do you determine when a limit exists or does not exist (DNE)?
3. What are some common pitfalls when evaluating limits?
4. How do limits apply to continuous functions?
5. What are the conditions for the existence of a limit?

**Tip:** Always double-check the point to which \(x\) is approaching. Small changes in the value \(x\) approaches can significantly affect the outcome.

Learn how to evaluate the limit as x approaches -7 of the expression (x^2 + 7x) / (x^2 - 2x - 63) with a detailed step-by-step solution. This problem covers key algebraic concepts, including limits, factoring quadratic equations, and properties of limits. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation