Math Problem Statement
Solution
The given mathematical expression is:
Solution:
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Let . Taking the natural logarithm on both sides:
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Now, consider the limit of :
Since grows slower than as , the numerator grows much slower than the denominator.
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Applying the growth comparison:
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Therefore:
Exponentiating back to get :
Final Answer:
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Here are 5 related questions for you to explore:
- How does the growth of compare to as ?
- Why is considered a slowly growing function?
- What are some common techniques to evaluate limits involving logarithmic functions?
- How do you use L'Hôpital's Rule to solve similar limits with as ?
- 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