7. \( \lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x^{2}+x-12} \)

To evaluate the limit: \[ \lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x^{2}+x-12} \] **Step 1: Substitute \( x = 3 \) to Check for Indeterminate Form** \[ \text{Numerator: } 3^{2} - 3 - 6 = 9 - 3 - 6 = 0 \] \[ \text{Denominator: } 3^{2} + 3 - 12 = 9 + 3 - 12 = 0 \] Since both the numerator and denominator evaluate to 0, we have an indeterminate form \( \frac{0}{0} \). This indicates that we can factor both the numerator and the denominator to simplify the expression. **Step 2: Factor the Numerator and Denominator** - **Numerator (\( x^2 - x - 6 \)):** We look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\). \[ x^2 - x - 6 = (x - 3)(x + 2) \] - **Denominator (\( x^2 + x - 12 \)):** We look for two numbers that multiply to \(-12\) and add to \(1\). These numbers are \(4\) and \(-3\). \[ x^2 + x - 12 = (x + 4)(x - 3) \] **Step 3: Simplify the Expression** Substitute the factored forms back into the limit: \[ \lim _{x \rightarrow 3} \frac{(x - 3)(x + 2)}{(x + 4)(x - 3)} \] Cancel the common factor \((x - 3)\): \[ \lim _{x \rightarrow 3} \frac{x + 2}{x + 4} \] **Step 4: Evaluate the Simplified Limit** Now, substitute \( x = 3 \) into the simplified expression: \[ \frac{3 + 2}{3 + 4} = \frac{5}{7} \] **Conclusion:** \[ \lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x^{2}+x-12} = \frac{5}{7} \] **Answer:** The limit is 5⁄7.