Determine the following limits.


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

Verified step by step guidance

1. Step 1: Identify the limit expression: \( \lim_{x \to 2^{+}} \frac{1}{\sqrt{x(x-2)}} \). This is a one-sided limit as \( x \) approaches 2 from the right.
2. Step 2: Analyze the behavior of the expression as \( x \to 2^{+} \). Note that \( x - 2 \) approaches 0 from the positive side, making the expression inside the square root approach 0.
3. Step 3: Consider the expression \( \sqrt{x(x-2)} \). As \( x \to 2^{+} \), \( x \) is slightly greater than 2, so \( x(x-2) \) is a small positive number, and \( \sqrt{x(x-2)} \) is also a small positive number.
4. Step 4: Evaluate the behavior of the entire fraction \( \frac{1}{\sqrt{x(x-2)}} \). As \( \sqrt{x(x-2)} \) approaches 0 from the positive side, the fraction approaches infinity.
5. Step 5: Conclude that the limit is \( +\infty \) as \( x \to 2^{+} \), since the denominator approaches 0 from the positive side, causing the fraction to grow without bound.

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, we are specifically looking at the limit as x approaches 2 from the right, which is denoted as x → 2⁺.

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. The limit from the right (denoted as x → c⁺) considers values greater than c, while the limit from the left (x → c⁻) considers values less than c. This distinction is crucial in this problem, as we are evaluating the limit as x approaches 2 from the right.

Square Root Function

The square root function, denoted as √x, is defined for non-negative values of x and is important in this limit problem. The expression under the square root, x(x - 2), must be non-negative for the limit to be defined. Understanding the behavior of the square root function near critical points, such as where the argument becomes zero, is essential for evaluating the limit correctly.