Math Problem Statement

lim (ln x)^(1/x) as x approaches infinity.

Solution

The given mathematical expression is:

limx(lnx)1x.\lim_{x \to \infty} (\ln x)^{\frac{1}{x}}.

Solution:

  1. Let y=(lnx)1xy = (\ln x)^{\frac{1}{x}}. Taking the natural logarithm on both sides: lny=1xln(lnx).\ln y = \frac{1}{x} \ln(\ln x).

  2. Now, consider the limit of lny\ln y: limxlny=limxln(lnx)x.\lim_{x \to \infty} \ln y = \lim_{x \to \infty} \frac{\ln(\ln x)}{x}.

    Since ln(lnx)\ln(\ln x) grows slower than xx as xx \to \infty, the numerator grows much slower than the denominator.

  3. Applying the growth comparison: ln(lnx)x0as x.\frac{\ln(\ln x)}{x} \to 0 \quad \text{as } x \to \infty.

  4. Therefore: lny0as x.\ln y \to 0 \quad \text{as } x \to \infty.

    Exponentiating back to get yy: ye0=1.y \to e^0 = 1.

Final Answer:

limx(lnx)1x=1.\lim_{x \to \infty} (\ln x)^{\frac{1}{x}} = 1.

Would you like a step-by-step breakdown or any clarifications? Let me know!


Here are 5 related questions for you to explore:

  1. How does the growth of lnx\ln x compare to xx as xx \to \infty?
  2. Why is ln(lnx)\ln(\ln x) considered a slowly growing function?
  3. What are some common techniques to evaluate limits involving logarithmic functions?
  4. How do you use L'Hôpital's Rule to solve similar limits with f(x)g(x)\frac{f(x)}{g(x)} as xx \to \infty?
  5. What is the significance of exponential functions in analyzing limits like this?

Tip: Always take the logarithm of an expression if it simplifies the exponents, especially in limits and growth rate comparisons.

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Math Problem Analysis

Mathematical Concepts

Limits
Logarithmic Functions
Growth Rate Analysis
Exponential Functions

Formulas

lim (ln x)^(1/x) as x -> ∞
ln y = (1/x) * ln(ln x)
lim (ln(ln x) / x) -> 0

Theorems

Logarithmic and Exponential Growth Comparison
Properties of Natural Logarithms

Suitable Grade Level

Grades 10-12