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 \(-\infty\).

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions near specific points, including points of discontinuity or indeterminate forms. Evaluating limits is crucial for defining derivatives and integrals.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In such cases, techniques like factoring, rationalizing, or applying L'Hôpital's Rule are used to resolve these forms and find the limit.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Understanding continuity is essential when evaluating limits, as it allows for the direct substitution of values in many cases, simplifying the limit evaluation process.