\( f ( x ) = \frac { x ^ { 2 } + 3 x + 2 } { x ^ { 2 } + 5 x + 4 } , x \rightarrow - 1 \)

To analyze the function \( f ( x ) = \frac { x ^ { 2 } + 3 x + 2 } { x ^ { 2 } + 5 x + 4 } \) as \( x \) approaches -1, we can start by factoring both the numerator and the denominator. The numerator \( x^2 + 3x + 2 \) factors to \( (x + 1)(x + 2) \) and the denominator \( x^2 + 5x + 4 \) factors to \( (x + 1)(x + 4) \). When substituting \( x = -1 \), both the numerator and denominator equal zero, indicating a potential indeterminate form. We can simplify the function by canceling the common factor \( (x + 1) \), leading to \( f(x) = \frac{x + 2}{x + 4} \) for \( x \neq -1 \). Now, we can easily evaluate the limit as \( x \) approaches -1: \[ \lim_{x \to -1} f(x) = \frac{-1 + 2}{-1 + 4} = \frac{1}{3}. \] So, the limit is \( \frac{1}{3} \). Simplifying expressions is a powerful technique in calculus! A common mistake is forgetting to factor correctly, leading to improper interpretations of limits, especially when facing indeterminate forms. Always double-check if you can simplify before substituting directly, as this can save time and prevent errors!