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Let's solve question 9b from the provided image:

The question asks for the following limit:

\[
\lim_{{x \to -1}} \frac{x^2 + 5x - 14}{x^2 - 4x + 4}
\]

### Step 1: Factor both the numerator and the denominator
First, factor both the numerator and the denominator:

- **Numerator:** \( x^2 + 5x - 14 \)
  - Find two numbers that multiply to \(-14\) and add up to \(5\): \(7\) and \(-2\).
  - The factored form is:
    \[
    x^2 + 5x - 14 = (x + 7)(x - 2)
    \]

- **Denominator:** \( x^2 - 4x + 4 \)
  - Recognize this as a perfect square trinomial:
    \[
    x^2 - 4x + 4 = (x - 2)^2
    \]

### Step 2: Substitute the factored forms into the limit expression
Now, rewrite the limit expression with the factored forms:
\[
\lim_{{x \to -1}} \frac{(x + 7)(x - 2)}{(x - 2)^2}
\]

### Step 3: Simplify the expression
Cancel out one factor of \((x - 2)\) from both the numerator and denominator:
\[
\lim_{{x \to -1}} \frac{x + 7}{x - 2}
\]

### Step 4: Evaluate the limit by substitution
Now, substitute \(x = -1\) into the simplified expression:
\[
\frac{-1 + 7}{-1 - 2} = \frac{6}{-3} = -2
\]

### Final Answer:
\[
\lim_{{x \to -1}} \frac{x^2 + 5x - 14}{x^2 - 4x + 4} = -2
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What happens if a limit expression results in an indeterminate form like \(\frac{0}{0}\)?
2. How do you factor a quadratic expression systematically?
3. What is the significance of cancelling terms when simplifying limits?
4. How would you approach this limit if it didn't factor easily?
5. What is the geometric interpretation of limits approaching specific values?

**Tip**: Always check if you can factor the expression before applying L'Hopital's Rule for limits involving indeterminate forms.

Solve 9b: Find the limit \(\lim_{{x \to -1}} \frac{x^2 + 5x - 14}{x^2 - 4x + 4}\).

This page explains how to solve the limit \(\lim_{{x \to -1}} \frac{x^2 + 5x - 14}{x^2 - 4x + 4}\) by factoring the quadratic equations. It includes step-by-step factoring of the numerator and denominator, simplifying the expression, and evaluating the limit.

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