Math Problem Statement
Solution
The problem involves evaluating the limit:
where and are positive real numbers and .
Solution:
We will approach this limit using L'Hôpital's rule and properties of logarithmic limits. The structure of the expression suggests that a logarithmic transformation will simplify the calculation.
Step-by-Step:
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Logarithmic Transformation:
Set the limit as:Taking the natural logarithm on both sides:
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Using L'Hôpital's Rule:
We encounter the form when evaluating the limit as , so we can apply L'Hôpital's rule.
= \lim_{t \to 0} \frac{\frac{d}{dt} \left[ \ln\left( \frac{b^{t+1} - a^{t+1}}{b - a} \right) \right]}{\frac{d}{dt}(t)}.$$ This simplifies to: $$\lim_{t \to 0} \frac{\frac{(b^{t+1} \ln b - a^{t+1} \ln a)}{b^{t+1} - a^{t+1}}}{1}.$$ -
Evaluating the Limit:
As , both and . Thus, the numerator becomes:
And the denominator approaches . Therefore, the limit can be written as:
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Using the Mean Value Theorem:
This limit is a well-known form, which is equal to .
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Exponential Back-Transformation:
Now, recall that:
Therefore:
Final Answer:
The value of the limit is:
Do you have any questions, or would you like me to explain a step further?
Here are 5 related questions for practice:
- How do you apply L'Hôpital's rule to logarithmic functions?
- What is the Mean Value Theorem, and how is it used in limits?
- What are some properties of exponential limits?
- How do logarithmic transformations simplify complex limits?
- Can you generalize this limit for more variables or different exponents?
Tip: Logarithmic transformations are powerful for handling exponential limits—they often turn multiplicative or power structures into additive forms for easier differentiation.
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Math Problem Analysis
Mathematical Concepts
Limits
Logarithmic Transformation
L'Hôpital's Rule
Mean Value Theorem
Formulas
L = lim (t→0) ((b^(t+1) - a^(t+1)) / (b - a))^(1/t)
ln(L) = lim (t→0) (1/t) ln((b^(t+1) - a^(t+1)) / (b - a))
lim (t→0) (b ln(b) - a ln(a)) / (b - a)
Theorems
L'Hôpital's Rule
Mean Value Theorem
Suitable Grade Level
Undergraduate Mathematics or Advanced High School Calculus
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