Determine the following limits.


b. $\underset{x\to 2^{-}}{\lim }\frac{1}{\sqrt{x(x-2)}}$

Verified step by step guidance

1. Identify the limit expression: \( \lim_{x \to 2^{-}} \frac{1}{\sqrt{x(x-2)}} \).
2. Recognize that as \( x \to 2^{-} \), the expression \( x - 2 \) approaches 0 from the negative side, making the expression inside the square root approach 0.
3. Consider the behavior of \( \sqrt{x(x-2)} \) as \( x \to 2^{-} \). Since \( x \) is slightly less than 2, \( x(x-2) \) is negative, and the square root of a negative number is not real.
4. Conclude that the expression inside the square root becomes undefined in the real number system as \( x \to 2^{-} \).
5. Determine that the limit does not exist in the real number system due to the square root of a negative number.

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 case, we are interested in the limit as x approaches 2 from the left, which is denoted as x → 2⁻.

Square Root Function

The square root function, denoted as √x, is a mathematical function that returns the non-negative value whose square is x. In the context of limits, the behavior of the square root function can significantly affect the limit's value, especially when the argument of the square root approaches zero, as it can lead to undefined or infinite values.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either from the left (denoted as x → a⁻) or from the right (denoted as x → a⁺). This concept is crucial when dealing with functions that may behave differently on either side of a point, such as in this limit problem where x approaches 2 from the left.