2. \( \lim _{x \rightarrow-3} \frac{x^{2}-2 x-3}{x^{2}-5 x+6}= \)

To find the limit \( \lim _{x \rightarrow-3} \frac{x^{2}-2 x-3}{x^{2}-5 x+6} \), we start by factoring both the numerator and denominator. The numerator can be factored as follows: \( x^{2}-2x-3 = (x-3)(x+1) \). The denominator factors as: \( x^{2}-5x+6 = (x-3)(x-2) \). This gives us the expression: \[ \lim _{x \rightarrow-3} \frac{(x-3)(x+1)}{(x-3)(x-2)}. \] Since \(x = -3\) does not make the denominator zero, we can cancel out the common factor \((x-3)\): \[ \lim _{x \rightarrow-3} \frac{x+1}{x-2}. \] Now, we can directly substitute \(x = -3\) into the simplified expression: \[ \frac{-3+1}{-3-2} = \frac{-2}{-5} = \frac{2}{5}. \] Therefore, the limit is \( \frac{2}{5} \).