The Value of (sqrt{ln x}) as (x) Approaches (e)

Understanding the behavior of mathematical functions as variables approach certain values is a fundamental concept in calculus and analysis. This article will explore the specific case of the function (sqrt{ln x}) as (x) approaches the natural number (e).

Introduction

Mathematically, determining the limit of a function as the variable approaches a specific value often requires careful handling of continuity and algebraic manipulation. In this context, we are interested in finding (lim_{x to e} sqrt{ln x}).

Case Analysis

First, let's consider the two-sided limit of the function (sqrt{ln x}) as (x) approaches (e).

Left-Hand Limit (LHL)

When (x) approaches (e) from the left, we can represent this as (lim_{x to e^-} sqrt{ln x}).

To visualize this, let (x e - epsilon), where (epsilon) is a small positive number. Therefore, as (epsilon to 0^ ), (x to e^- ). Now, let's substitute (b epsilon) to reframe the limit in a familiar form.

For small values of (b), (ln(e - b) approx ln(1 - frac{b}{e})). Since (ln(1 - b) approx -b) for small (b), we can simplify (ln(e - b) approx -frac{b}{e}).

So, (sqrt{ln(e - b)} approx sqrt{-frac{b}{e}}), and as (b to 0), (sqrt{-frac{b}{e}} to 1). Thus, the left-hand limit is:

[lim_{x to e^-} sqrt{ln x} 1]

Right-Hand Limit (RHL)

Similarly, when (x) approaches (e) from the right, we represent this as (lim_{x to e^ } sqrt{ln x}).

Let (x e epsilon), where (epsilon) is a small positive number. Therefore, as (epsilon to 0^ ), (x to e^ ). Now, let's substitute (b epsilon) to reframe the limit in a familiar form.

For small values of (b), (ln(e b) approx ln(1 frac{b}{e})). Since (ln(1 b) approx frac{b}{e}) for small (b), we can simplify (ln(e b) approx frac{b}{e}).

So, (sqrt{ln(e b)} approx sqrt{frac{b}{e}}), and as (b to 0), (sqrt{frac{b}{e}} to 1). Thus, the right-hand limit is:

[lim_{x to e^ } sqrt{ln x} 1]

Combining Both Limits

Since both the left-hand limit and the right-hand limit as (x) approaches (e) are equal to 1, we can conclude that the limit as (x) approaches (e) from either side is 1.

Alternative Approach: Function Continuity

We can also use the fact that the logarithmic and square root functions are continuous. Therefore, (sqrt{ln x}) is continuous for (x > 0) and (x eq 1). This means that:

[lim_{x to e} sqrt{ln x} sqrt{ln e} sqrt{1} 1]

Conclusion

The value of (sqrt{ln x}) as (x) approaches (e) is 1.

Key Concepts

Limit: The value that a function approaches as the input approaches a certain value. Natural Logarithm: The logarithm to the base (e). Square Root: The inverse operation of squaring a number.

Understanding these key concepts and their interplay is crucial for solving such problems in calculus.