Determine the following limits.


a. $\underset{x\to 1^{+}}{\lim }\frac{x-3}{\sqrt{x^2-5x+4}}$

Verified step by step guidance

1. Identify the limit expression: \( \lim_{x \to 1^{+}} \frac{x-3}{\sqrt{x^2-5x+4}} \).
2. Factor the expression under the square root: \( x^2 - 5x + 4 = (x-1)(x-4) \).
3. Substitute \( x = 1^{+} \) into the factored expression: \( \sqrt{(x-1)(x-4)} \) becomes \( \sqrt{(1^{+}-1)(1^{+}-4)} \).
4. Analyze the behavior of the numerator and denominator as \( x \to 1^{+} \): The numerator \( x-3 \) approaches \( 1-3 = -2 \), and the denominator approaches \( \sqrt{0 \cdot (-3)} = 0 \).
5. Conclude that the limit involves division by zero, indicating a potential vertical asymptote or undefined behavior, and further analysis is needed to determine the limit's behavior.

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near points of interest, including points where they may not be defined. In this question, we are specifically looking at the limit as x approaches 1 from the right, which is denoted as x → 1⁺.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given limit, the numerator is a linear polynomial (x - 3) and the denominator involves a square root of a quadratic polynomial (√(x² - 5x + 4)). Understanding how to simplify and analyze rational functions is crucial for evaluating limits, especially when approaching points that may lead to indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when the limit of a function results in an ambiguous expression, such as 0/0 or ∞/∞. In this case, as x approaches 1, both the numerator and the denominator approach specific values that may lead to an indeterminate form. Recognizing and resolving these forms, often through algebraic manipulation or L'Hôpital's rule, is essential for finding the correct limit.