Determine the sollowing limits and explain your reasoning. a) \( \lim _{x \rightarrow-3} \frac{x^{2}-4 x+3}{x^{2}+3 x+2} \)

To determine the limit \[ \lim _{x \rightarrow -3} \frac{x^{2}-4x+3}{x^{2}+3x+2}, \] we will follow these steps: 1. **Substitute the value of \(x\)**: First, we will substitute \(x = -3\) directly into the expression to see if we can evaluate the limit directly. 2. **Factor the numerator and denominator**: If direct substitution results in an indeterminate form (like \(\frac{0}{0}\)), we will factor both the numerator and the denominator to simplify the expression. 3. **Re-evaluate the limit**: After simplification, we will re-evaluate the limit. Let's start with step 1: ### Step 1: Direct Substitution Substituting \(x = -3\): - Numerator: \[ (-3)^{2} - 4(-3) + 3 = 9 + 12 + 3 = 24. \] - Denominator: \[ (-3)^{2} + 3(-3) + 2 = 9 - 9 + 2 = 2. \] So, we have: \[ \frac{24}{2} = 12. \] Since we did not get an indeterminate form, we can conclude that: \[ \lim _{x \rightarrow -3} \frac{x^{2}-4x+3}{x^{2}+3x+2} = 12. \] ### Conclusion The limit is \[ \boxed{12}. \] This result was obtained by direct substitution, confirming that the function is continuous at \(x = -3\).