Problem 55 - University Calculus: Early Transcendentals

Question

Find the limits. $$\lim _{x \rightarrow 0^{+}} x^{-1 / \ln x}$$

Step-by-Step Solution

Answer

The limit is \( e^{-1} \).

Step 1: Recognize the Indeterminate Form

Observe that as \( x \rightarrow 0^+ \), both \( x^{-1 / \ln x} \) and \( \ln x \) approach indeterminate forms. \( \ln x \rightarrow -\infty \) and \( -1 / \ln x \rightarrow 0 \), making \( x^{-1/\ln x} = e^{\ln(x^{-1/\ln x})} \) an indeterminate form of type \( 0^0 \).

Step 2: Use Logarithmic Transformation

Set \( y = x^{-1/\ln x} \), and take the natural logarithm of both sides: \( \ln y = -\frac{1}{\ln x} \cdot \ln x = -\frac{\ln x}{\ln x} = -1 \). This simplification reveals a constant in the exponent.

Step 3: Analyze the Exponential Form

Since \( \ln y = -1 \), then \( y = e^{-1} \) as \( x \rightarrow 0^+ \). Therefore, the limit of the original expression \( x^{-1/\ln x} \) is determined by noting that it converges to \( e^{-1} \).

Step 4: Conclusion about the Limit

Thus, the limit \( \lim_{x \rightarrow 0^+} x^{-1/\ln x} \) is equal to \( e^{-1} \). This solves the initial indeterminate form.

Key Concepts

Indeterminate Forms

When we talk about indeterminate forms in calculus, we're referring to expressions that do not have a clear limit or value at first glance. These forms typically appear in limit problems, such as when evaluating the limit of a function as it approaches a certain point where it is not initially clear what the outcome will be. Common examples include forms like \(0/0\), \(\infty/\infty\), \(0^0\), and several others.
When you encounter an indeterminate form, it requires further analysis or transformation to simplify or resolve the expression. In our exercise, as \( x \to 0^+ \), \( x^{-1/\ln x} \) needs to be carefully examined due to the presence of the \(0^0\) form. Tackling such forms often involves using further techniques such as algebraic manipulation or applying the properties of logarithms and exponents. Recognizing and dealing with indeterminate forms is essential in understanding limits and the behavior of functions near specific points.

Logarithmic Transformation

Logarithmic transformations are powerful tools for simplifying complex expressions, especially when exponents are involved. In this exercise, a logarithmic transformation is used to simplify \( x^{-1/\ln x} \). By introducing a new variable \( y = x^{-1/\ln x} \) and considering its natural logarithm, we convert the expression into a form that reveals its true nature.
By taking the natural logarithm, we turn the exponential expression into a product: \( \ln y = -\frac{\ln x}{\ln x} = -1 \). This conversion drastically simplifies the problem by translating the indeterminate form into something manageable.

• Logarithms help reduce the complexity of problems and can convert multiplication into addition, division into subtraction, and exponents into coefficients.
• In calculus, logarithmic transformations can turn difficult limit problems into simple algebraic manipulations.

Understanding this process is vital, as it allows you to see complex expressions in a different light and find solutions more intuitively.

Exponential Growth

Exponential functions, characterized by expressions of the form \( e^{x} \), play a crucial role in modeling growth and decay processes in nature. However, when used in calculus to find limits, they often reveal behaviors of functions as variables approach certain points.
In our example, we analyze the behavior of \( x^{-1/\ln x} \) which elegantly resolves into an exponential form \( e^{-1} \) by using logarithms to simplify and reveal the constant hidden in the expression.
This transformation shows that as \( x \to 0^+ \), the limit isn't some elusive indeterminate result but rather becomes a well-defined number \( e^{-1} \), demonstrating the sophisticated power of exponential calculations.

• When faced with such functions, recognizing transformations that reveal a simpler exponential relationship is key.
• Exponential growth and decay aren't just mathematical curiosities; they are fundamental behaviors encountered across different scientific fields.
• Grasping how to manage exponential terms in limit problems will give you deeper insights into the seamless transition between apparent complexity and underlying simplicity.

Mastering these concepts allows for a greater understanding of how functions behave in critical scenarios.