Determine the following limits.


a. lim x→2^+ x^2 − 4x + 3 / (x − 2)^2

Verified step by step guidance

1. Step 1: Identify the type of limit problem. This is a one-sided limit as \( x \) approaches 2 from the right (\( x \to 2^+ \)).
2. Step 2: Substitute \( x = 2 \) into the function \( \frac{x^2 - 4x + 3}{(x - 2)^2} \) to check if it results in an indeterminate form. Substituting gives \( \frac{2^2 - 4 \times 2 + 3}{(2 - 2)^2} = \frac{4 - 8 + 3}{0} = \frac{-1}{0} \), indicating a division by zero.
3. Step 3: Analyze the behavior of the numerator and denominator as \( x \to 2^+ \). The numerator \( x^2 - 4x + 3 \) simplifies to \( (x - 1)(x - 3) \). As \( x \to 2^+ \), \( x - 1 \to 1 \) and \( x - 3 \to -1 \), so the numerator approaches \( 1 \times -1 = -1 \).
4. Step 4: Consider the denominator \( (x - 2)^2 \). As \( x \to 2^+ \), \( x - 2 \to 0^+ \), so \( (x - 2)^2 \to 0^+ \).
5. Step 5: Determine the limit by combining the behavior of the numerator and denominator. Since the numerator approaches \(-1\) and the denominator approaches \(0^+\), the limit approaches \(-\infty\).