Understanding the Limit of x1/√x / (x1/√x - x1/√x) as x Approaches Infinity

In the field of calculus, understanding limits is fundamental. This article focuses on a specific limit problem and provides a detailed explanation of the steps involved to resolve it. We will explore the mathematical reasoning behind the limit of the expression "x^{1/sqrt{x}} / (x^{1/sqrt{x}} - x^{1/sqrt{x}}-1)" as x approaches infinity. The solution reveals that the limit is infinity, but the detailed steps are crucial for a comprehensive understanding.

Step-by-Step Solution to the Limit Problem

Let's begin by dividing the numerator and denominator of the expression by . The goal is to simplify the fraction and make it easier to analyze the behavior as x approaches infinity.

After this division, the expression becomes:

"1 / (1 - x^{1/sqrt{x}})

Now, let's focus on the term . To simplify further, let's use a substitution: . This transforms into . The expression now becomes:

"1 / (1 - y^{2/y})

For simplicity, the term can be further analyzed. We start by examining the logarithm:

"log(y^{2/y} - 1/y)

This expression can be simplified by using the properties of logarithms:

"log(y) log((1 (1/y)^2 - 1/y)^{1/y})

By further expansion, we find that the term inside the logarithm simplifies to a form that tends to zero as y approaches infinity (or x approaches infinity). This is because the logarithm of a term that approaches one (through a series of simplifications and approximations) tends to zero.

Therefore:

"e^0 1

As the denominator of the expression tends to zero, the entire expression tends to infinity. Hence the limit is:

"lim_{xtoinfty}x^{1/sqrt{x}} / (x^{1/sqrt{x}} - 1) infty

Key Takeaways

Substitution: Using a substitution such as "y sqrt{x} Logarithms: Simplifying expressions using logarithms can help in understanding the behavior of functions as x approaches infinity. Limit Analysis: Carefully analyzing the behavior of terms as they approach infinity is essential for solving complex limit problems.

Conclusion

Understanding the limit of "x^{1/sqrt{x}} / (x^{1/sqrt{x}} - 1)" as x approaches infinity involves a series of steps that include substitution, logarithmic simplification, and careful analysis. This understanding not only helps in solving the specific problem but also reinforces fundamental concepts in calculus.

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