Determine the following limits.
lim x→3 1/ x − 3(1 /√x + 1 − 1/2)

Verified step by step guidance

1. Identify the limit expression: \( \lim_{{x \to 3}} \left( \frac{1}{x - 3} \left( \frac{1}{\sqrt{x + 1}} - \frac{1}{2} \right) \right) \).
2. Notice that direct substitution of \( x = 3 \) results in an indeterminate form \( \frac{0}{0} \).
3. To resolve the indeterminate form, consider simplifying the expression inside the limit by finding a common denominator for the terms inside the parentheses.
4. The common denominator for \( \frac{1}{\sqrt{x + 1}} \) and \( \frac{1}{2} \) is \( 2\sqrt{x + 1} \). Rewrite the expression as \( \frac{2 - \sqrt{x + 1}}{2\sqrt{x + 1}} \).
5. Multiply the numerator and the denominator by the conjugate of the numerator, \( 2 + \sqrt{x + 1} \), to rationalize the expression and simplify further.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this question, evaluating the limit as x approaches 3 is crucial for determining the function's value at that point.

Indeterminate Forms

Indeterminate forms occur in calculus when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is essential for applying techniques such as L'Hôpital's Rule or algebraic manipulation to resolve the limit. In this case, the expression may lead to an indeterminate form, necessitating further analysis.

Rational Functions

Rational functions are ratios of polynomials, and their limits can often be evaluated by simplifying the expression. Understanding how to manipulate these functions, including factoring and canceling common terms, is vital for finding limits. In the given limit problem, recognizing the structure of the rational function will aid in simplifying the expression before evaluating the limit.