We are tasked with finding the limit of the following expression as \( x \to -1 \):

\[
\lim_{{x \to -1}} \frac{-2x - 2}{x^2 - 6x - 7}
\]

### Step 1: Evaluate the function at \( x = -1 \)
First, substitute \( x = -1 \) into the expression:

\[
\text{Numerator: } -2(-1) - 2 = 2 - 2 = 0
\]
\[
\text{Denominator: } (-1)^2 - 6(-1) - 7 = 1 + 6 - 7 = 0
\]

Since both the numerator and denominator equal zero, we have an indeterminate form \( \frac{0}{0} \). Therefore, we need to simplify the expression further, usually by factoring.

### Step 2: Factor the numerator and denominator
- The numerator \( -2x - 2 \) can be factored as:
  \[
  -2x - 2 = -2(x + 1)
  \]
- The denominator \( x^2 - 6x - 7 \) can be factored by finding two numbers that multiply to \(-7\) and add to \(-6\). These numbers are \(-7\) and \(1\), so:
  \[
  x^2 - 6x - 7 = (x - 7)(x + 1)
  \]

### Step 3: Simplify the expression
Now substitute the factored forms back into the limit:
\[
\lim_{{x \to -1}} \frac{-2(x + 1)}{(x - 7)(x + 1)}
\]

The common factor \( (x + 1) \) cancels out:
\[
\lim_{{x \to -1}} \frac{-2}{x - 7}
\]

### Step 4: Evaluate the simplified limit
Now, substitute \( x = -1 \) into the simplified expression:
\[
\frac{-2}{-1 - 7} = \frac{-2}{-8} = \frac{1}{4}
\]

### Final Answer:
\[
\lim_{{x \to -1}} \frac{-2x - 2}{x^2 - 6x - 7} = \frac{1}{4}
\]

Would you like further details or have any questions?

Here are five related questions you might find helpful:
1. How do you determine if a limit results in an indeterminate form?
2. What are other common methods for resolving indeterminate forms besides factoring?
3. How do you find the limit as \( x \to \infty \)?
4. Can L'Hopital's Rule be applied here? If so, how would it work?
5. How do you factor a quadratic equation efficiently?

**Tip:** Always check for common factors in both the numerator and denominator to simplify limits involving indeterminate forms.

Explore how to solve a limit problem involving an indeterminate form and factoring. This step-by-step guide helps students understand the limit as x approaches -1 for the expression (-2x - 2)/(x^2 - 6x - 7). Learn how to factor both the numerator and denominator to simplify and solve the limit. Suitable for Grades 11-12 and students learning advanced algebra or calculus concepts.

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