Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).
f(x) = (x² − 4x + 3) / (x − 1)

Verified step by step guidance

1. Step 1: Identify the points where the denominator is zero, as these are potential vertical asymptotes. Set the denominator equal to zero: x - 1 = 0, which gives x = 1.
2. Step 2: Confirm that x = 1 is a vertical asymptote by checking that the numerator does not also become zero at x = 1. Substitute x = 1 into the numerator: (1)^2 - 4(1) + 3 = 0, which means the numerator is zero, so x = 1 is not a vertical asymptote but a removable discontinuity.
3. Step 3: Factor the numerator to simplify the expression. The numerator x^2 - 4x + 3 can be factored as (x - 1)(x - 3).
4. Step 4: Simplify the function by canceling the common factor (x - 1) from the numerator and the denominator, resulting in f(x) = x - 3 for x ≠ 1.
5. Step 5: Since the factor (x - 1) cancels out, there are no vertical asymptotes. However, analyze the behavior around x = 1 to understand the nature of the discontinuity.