Exploring the Limit of the Function -√(1 - x^2) as x Approaches -1 from the Left

In this article, we will delve into the limit of the function -√(1 - x^2) as x approaches -1 from the left. We will explore the domain, evaluate the limit, and discuss why the limit does not exist in some cases while it exists in others.

Domain of -√(1 - x^2)

The function -√(1 - x^2) is defined for all real numbers x such that the expression inside the square root is non-negative. This means we need to find the values of x for which 1 - x^2 ≥ 0. Solving the inequality:

1 - x^2 ≥ 0

x^2 ≤ 1

-1 ≤ x ≤ 1

Therefore, the domain of the function is the interval [-1, 1].

Limit as x Approaches -1 from the Left

To find the limit of the function -√(1 - x^2) as x approaches -1 from the left, we need to consider the behavior of the function as x gets closer and closer to -1 from values smaller than -1.

Mathematically, we express this as:

limx→-1- -√(1 - x^2)

Let's evaluate this limit step by step.

Step 1: Determine the Expression Inside the Square Root

The expression inside the square root is:

-1 - h

Where h is a small negative number approaching 0 from the left. As h gets closer to 0, -1 - h gets closer to -1 from the left. Substituting -1 - h into the function:

-√(1 - (-1 - h)^2)

Simplifying the expression inside the square root:

-√(1 - (1 2h h^2))

-√(1 - 1 - 2h - h^2)

-√(-2h - h^2)

This expression is undefined for real numbers because the expression under the square root is negative. Therefore, the limit is not defined for real numbers as x approaches -1 from the left.

Step 2: Imaginary Axis Approach

When approaching -1 from the left, we can consider the limit in the complex plane. If we let x -1 - h where h is a small positive number approaching 0, then:

-√(1 - (-1 - h)^2) -√(1 - (1 2h h^2)) -√(-2h - h^2)

This expression can be written in terms of imaginary numbers. The expression under the square root is:

-2h - h^2 -h(2 h)

The square root of a negative number is imaginary, so we can write:

-√(-h(2 h)) -√(-h) √(2 h) -i√h √(2 h)

As h approaches 0, the term -i√h approaches 0, and √(2 h) approaches √2. Therefore, the limit approaches 0.

Conclusion

To summarize, the function -√(1 - x^2) is defined for all x in the interval [-1, 1]. When evaluating the limit as x approaches -1 from the left, the function is undefined for real numbers but approaches 0 in the complex plane.

Keywords: limit, function, square root