Math Problem Statement
Solution
To evaluate the limit
we'll analyze the behavior of the function as approaches infinity.
Step 1: Apply L'Hôpital's Rule
Since both and grow without bounds as , we have an indeterminate form of , which allows us to apply L'Hôpital's Rule. This rule states that for functions of the form where both and as , we can find the limit by differentiating the numerator and the denominator.
Step 2: Differentiate the Numerator and the Denominator
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Differentiate the numerator, :
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Differentiate the denominator, :
Step 3: Rewrite the Limit Using the Derivatives
The limit now becomes:
Step 4: Simplify the Expression
Rewrite as:
Step 5: Take the Limit
Now, as , .
Therefore:
Final Answer
Would you like more details or have any questions?
Here are five related questions for further exploration:
- What is L'Hôpital's Rule and when can it be applied?
- How does the growth rate of compare to for various values of ?
- Can we generalize this limit if the denominator is changed to for some ?
- What are other methods to analyze limits of the form as ?
- How does the behavior of logarithmic functions compare to polynomial and exponential functions as approaches infinity?
Tip: For limits involving and powers of , it can often help to compare their growth rates, as polynomial functions tend to grow faster than logarithmic functions.
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Math Problem Analysis
Mathematical Concepts
Calculus
Limits
Logarithmic and Radical Functions
Formulas
L'Hôpital's Rule
Differentiation rules for \( \ln x \) and \( x^{1/2} \)
Theorems
L'Hôpital's Rule
Suitable Grade Level
Undergraduate Calculus
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