Evaluate lim x→2^+ √x−2.

Verified step by step guidance

1. Identify the type of limit: This is a one-sided limit as x approaches 2 from the right (x→2^+).
2. Understand the behavior of the function: As x approaches 2 from the right, x is slightly greater than 2, so x - 2 is a small positive number.
3. Consider the expression under the square root: Since x is approaching 2 from the right, the expression \( \sqrt{x - 2} \) involves taking the square root of a small positive number.
4. Analyze the limit: As x gets closer to 2 from the right, the value of \( x - 2 \) approaches 0, making \( \sqrt{x - 2} \) approach the square root of 0.
5. Conclude the behavior of the limit: The limit of \( \sqrt{x - 2} \) as x approaches 2 from the right is the square root of 0.

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. In this case, we are interested in the limit as x approaches 2 from the right (denoted as 2^+). Understanding limits is crucial for analyzing the continuity and behavior of functions at specific points.

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 notation lim x→2^+ indicates that we are considering values of x that are greater than 2. This concept is important for understanding how functions behave near points of discontinuity or where they may not be defined.

Square Root Function

The square root function, denoted as √x, is a function that returns the non-negative value whose square is x. In the context of the limit, we are evaluating √x - 2 as x approaches 2. Understanding the properties of the square root function, including its domain and behavior near specific points, is essential for accurately calculating the limit.