Questions: lim as x approaches 3 of (x-3)/(x^2-4x+3) =

Solution Steps

To solve the limit problem, we need to simplify the expression. First, factor the denominator. Then, cancel out the common factors in the numerator and the denominator. Finally, substitute the value of \( x \) approaching 3 into the simplified expression to find the limit.

We start with the limit expression: \[ \lim_{x \rightarrow 3} \frac{x - 3}{x^2 - 4x + 3} \] The denominator can be factored as follows: \[ x^2 - 4x + 3 = (x - 1)(x - 3) \] Thus, we can rewrite the limit as: \[ \lim_{x \rightarrow 3} \frac{x - 3}{(x - 1)(x - 3)} \]

Next, we cancel the common factor \(x - 3\) from the numerator and the denominator: \[ \lim_{x \rightarrow 3} \frac{1}{x - 1} \]

Now, we substitute \(x = 3\) into the simplified expression: \[ \frac{1}{3 - 1} = \frac{1}{2} \]

Final Answer