Math Problem Statement

Suppose that a and b (b ≠ a) are real positive numbers. Then the value of lim (t→0) ((b^(t+1) - a^(t+1)) / (b - a))^(1/t) is equal to?

Solution

The problem involves evaluating the limit:

limt0(bt+1at+1ba)1t,\lim_{t \to 0} \left( \frac{b^{t+1} - a^{t+1}}{b - a} \right)^{\frac{1}{t}},

where aa and bb are positive real numbers and aba \neq b.


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:

  1. Logarithmic Transformation:
    Set the limit as:

    L=limt0(bt+1at+1ba)1t.L = \lim_{t \to 0} \left( \frac{b^{t+1} - a^{t+1}}{b - a} \right)^{\frac{1}{t}}.

    Taking the natural logarithm on both sides:

    ln(L)=limt01tln(bt+1at+1ba).\ln(L) = \lim_{t \to 0} \frac{1}{t} \ln\left( \frac{b^{t+1} - a^{t+1}}{b - a} \right).

  2. Using L'Hôpital's Rule:

    We encounter the form 00\frac{0}{0} when evaluating the limit as t0t \to 0, 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}.$$
  3. Evaluating the Limit:

    As t0t \to 0, both bt+1bb^{t+1} \to b and at+1aa^{t+1} \to a. Thus, the numerator becomes:

    blnbalna.b \ln b - a \ln a.

    And the denominator approaches bab - a. Therefore, the limit can be written as:

    limt0blnbalnaba.\lim_{t \to 0} \frac{b \ln b - a \ln a}{b - a}.

  4. Using the Mean Value Theorem:

    This limit is a well-known form, which is equal to ln(ba)\ln \left( \frac{b}{a} \right).

  5. Exponential Back-Transformation:

    Now, recall that:

    ln(L)=ln(ba).\ln(L) = \ln\left( \frac{b}{a} \right).

    Therefore:

    L=ba.L = \frac{b}{a}.


Final Answer:

The value of the limit is:

ba.\boxed{\frac{b}{a}}.


Do you have any questions, or would you like me to explain a step further?

Here are 5 related questions for practice:

  1. How do you apply L'Hôpital's rule to logarithmic functions?
  2. What is the Mean Value Theorem, and how is it used in limits?
  3. What are some properties of exponential limits?
  4. How do logarithmic transformations simplify complex limits?
  5. 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