Evaluate \(\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2}\)

Answer Details

To evaluate the limit \(\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2}\), we can start by substituting \(x = 1\) directly into the expression. However, this results in an expression of the form \(\frac{0}{0}\), which is undefined. To evaluate such a limit, we can use algebraic techniques to simplify the expression before taking the limit. In this case, we can factor the denominator to obtain: \[\frac{1 - x}{x^{2} - 3x + 2} = \frac{1 - x}{(x - 1)(x - 2)} = -\frac{1}{x - 2}\] Now, we can take the limit as \(x\) approaches 1 by substituting \(x = 1\) directly into the simplified expression. This gives: \[\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2} = \lim \limits_{x \to 1} -\frac{1}{x - 2} = -\frac{1}{-1} = 1\] Therefore, the limit of the expression as \(x\) approaches 1 is equal to 1. So, the correct option is 1.