Find the limit as x approaches -5, ((x^2)-2x-35)/((x^2)+11x+30)

We are tasked with finding the limit:

\[
\lim_{x \to -5} \frac{x^2 - 2x - 35}{x^2 + 11x + 30}
\]

### Step 1: Factor both the numerator and the denominator
First, let's factor the expressions in both the numerator and the denominator.

#### Numerator: \( x^2 - 2x - 35 \)
We need two numbers that multiply to \(-35\) and add to \(-2\). These numbers are \(-7\) and \(5\), so the factorization is:
\[
x^2 - 2x - 35 = (x - 7)(x + 5)
\]

#### Denominator: \( x^2 + 11x + 30 \)
We need two numbers that multiply to \(30\) and add to \(11\). These numbers are \(5\) and \(6\), so the factorization is:
\[
x^2 + 11x + 30 = (x + 5)(x + 6)
\]

### Step 2: Rewrite the expression
Now we substitute the factored forms into the original limit expression:
\[
\lim_{x \to -5} \frac{(x - 7)(x + 5)}{(x + 5)(x + 6)}
\]

### Step 3: Simplify the expression
We can cancel out the common factor \((x + 5)\) from the numerator and the denominator, provided \(x \neq -5\):
\[
\lim_{x \to -5} \frac{x - 7}{x + 6}
\]

### Step 4: Substitute \(x = -5\) into the simplified expression
Now, we substitute \(x = -5\) into the simplified expression:
\[
\frac{-5 - 7}{-5 + 6} = \frac{-12}{1} = -12
\]

Thus, the limit is:
\[
\boxed{-12}
\]

Would you like further details or have any questions? Here are five related questions you might find interesting:

1. What happens if we don't cancel the common factor and try direct substitution from the beginning?
2. How would the limit change if the numerator and denominator didn't have a common factor?
3. How do limits behave at infinity for rational functions?
4. What is the general strategy for finding limits of rational functions?
5. How do removable and non-removable discontinuities affect limits?

**Tip**: Always check if factoring is possible before applying direct substitution for limits of rational functions.

This guide explains how to calculate the limit of a rational function as x approaches -5 using factoring and simplification techniques. The problem involves factoring quadratic equations and applying limit laws, suitable for students in Grades 10-12.

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