Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(c_{1}, c_{2}, \ldots, c_{n}\) be positive constants with \(\sum_{i=1}^{n} c_{i}=1\), and let \(x_{1}, x_{2}, \ldots, x_{n}\) be positive numbers. Take natural logarithms and then use l'Hôpital's Rule to show that $$ \lim _{t \rightarrow 0^{+}}\left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1 / t}=x_{1}^{c_{1}} x_{2}^{c_{2} \cdots} x_{n}^{c_{n}}=\prod_{i=1}^{n} x_{i}^{c_{i}} $$ Here \(\prod\) means product; that is, \(\prod_{i=1}^{n} a_{i}\) means \(a_{1} \cdot a_{2} \cdots \cdots a_{n}\). In particular, if \(a, b, x\), and \(y\) are positive and \(a+b=1\), then $$ \lim _{t \rightarrow 0^{+}}\left(a x^{t}+b y^{t}\right)^{1 / t}=x^{a} y^{b} $$

Step-by-Step Solution

Answer

The limit equals \( \prod_{i=1}^{n} x_{i}^{c_{i}} \).

Step 1: Consider the Expression Inside the Limit

Given the expression \( \left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1/t} \), rewrite it using the natural logarithm: \( \ln\left(\left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1/t}\right) = \frac{1}{t} \ln\left( \sum_{i=1}^{n} c_{i} x_{i}^{t} \right) \). We aim to find \( \lim_{t \to 0^+} \ln\left(\left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1 / t}\right) \).

Step 2: Apply l'Hôpital's Rule to the Natural Log

Set \( L(t) = \ln\left( \sum_{i=1}^{n} c_{i} x_{i}^{t} \right) \). Our aim is to find \( \lim_{t \to 0^+} \frac{L(t)}{t} \). As \( t \to 0^+ \), the expression has a \(0/0\) form, allowing us to use l'Hôpital's Rule.

Step 3: Differentiate the Numerator and Denominator

Differentiate the numerator \( L(t) \) using the derivative of natural logarithm: \( L'(t) = \frac{d}{dt}\ln \left( \sum_{i=1}^{n} c_{i} x_{i}^{t} \right) = \frac{\sum_{i=1}^{n} c_{i} x_{i}^{t} \ln(x_{i})}{\sum_{i=1}^{n} c_{i} x_{i}^{t}} \).The derivative of the denominator \( t \) is simply \( 1 \).

Step 4: Simplify the Limit using Differentiated Expression

Compute the limit: \[\lim_{t \to 0^+} \frac{\sum_{i=1}^{n} c_{i} x_{i}^{t} \ln(x_{i})}{\sum_{i=1}^{n} c_{i} x_{i}^{t}} = \lim_{t \to 0^+} \sum_{i=1}^{n} c_{i} \ln(x_{i}) \cdot \frac{x_{i}^{t}}{\sum_{i=1}^{n} c_{i} x_{i}^{t}}.\]As \(t \to 0^+\), each term \( x_{i}^{t} \to 1 \), hence the limit evaluates to \( \sum_{i=1}^{n} c_{i} \ln(x_{i}) \).

Step 5: Exponentiate the Result

Since we initially took a natural logarithm, we'll exponentiate the limit result: \( \lim_{t \to 0^+} \frac{L(t)}{t} = \sum_{i=1}^{n} c_{i} \ln(x_{i}) \). Then, the original limit becomes: \[\lim_{t \to 0^+} \left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1/t} = e^{\sum_{i=1}^{n} c_{i} \ln(x_{i})}.\]

Step 6: Reshape the Exponential Expression

Utilize the property of logarithms and exponentials: \( e^{\sum_{i=1}^{n} c_{i} \ln(x_{i})} = \prod_{i=1}^{n} e^{c_{i} \ln(x_{i})} = \prod_{i=1}^{n} x_{i}^{c_{i}} \). This verifies the expression matches the formula \( x_{1}^{c_{1}} x_{2}^{c_{2}} \cdots x_{n}^{c_{n}} = \prod_{i=1}^{n} x_{i}^{c_{i}} \).

Key Concepts

Limit Evaluation

In calculus, evaluating limits is an essential technique used to understand the behavior of functions as they approach certain points. When we talk about limits in the context of this exercise, we are trying to find what the function \[ \left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1/t} \] behaves like as the variable \( t \) approaches zero from the positive side, indicated by \( t \to 0^+ \).

In this case, the expression originally gives a form that is not immediately defined, specifically a \( 0/0 \) form, which necessitates the use of techniques like l'Hôpital's Rule to resolve. This rule allows you to evaluate limits of indeterminate forms \((0/0)\) by differentiating the numerator and the denominator separately.

The evaluated limit \[ \lim_{t\to0^+} \left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1/t} \] results in a well-defined mathematical form recognized as the product notation or the weighted geometric mean in this context.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a way to express growth and decay processes, especially in the context of exponential functions. It is the inverse of the exponential function \(e^x\) and plays a critical role in calculus and limit evaluations.

In this exercise, natural logarithms are used to convert the complex product into a more manageable sum, making the application of l'Hôpital's Rule feasible. By taking the natural logarithm of a function, it's easier to handle and differentiate during the limit evaluation process.

The key properties of the natural logarithm that assist in simplifying calculations include:

• \(\ln(ab) = \ln a + \ln b\)
• \(\ln(a^b) = b \ln a\)
• The derivative \( \frac{d}{dt} \ln u = \frac{1}{u} \frac{du}{dt} \)

Thus, the natural logarithm transforms the multiplication operations inside the limit into addition, allowing for differentiation in the limit evaluation.

Product Notation

Product notation, symbolized by \( \prod \), is a compact way to indicate the multiplication of a series of terms. In calculus and algebra, it provides a concise way to write lengthy multiplication operations and is particularly handy when dealing with expressions involving large numbers of terms.

In the context of the given problem, the product notation describes the weighted geometric mean:\[ \prod_{i=1}^{n} x_{i}^{c_{i}} = x_{1}^{c_{1}} x_{2}^{c_{2}} \cdots x_{n}^{c_{n}} \]This is a generalization of the exponential growth rate, where each \( x_{i} \) is raised to the power of its corresponding weight \( c_{i} \).

The product notation not only simplifies the expression but also provides clarity in mathematical derivations, particularly where multiple multiplicative components are involved. It's a key element in translating the limit problem into a more understandable analytical form, showcasing how powers and weights interact smoothly within the scope of calculus.

Exponential Function

The exponential function, \(e^x\), is fundamental in mathematics, representing continuous growth or decay. It’s closely linked with the natural logarithm function, which is its inverse. Together, these functions are powerful tools in calculus for solving problems involving rates of change and limits.

In the exercise, exponentiation is the last step after completing the limit evaluation. Originally, natural logarithms simplify expressions under the limit; then, the solution requires exponentiating back from the logarithm form to the exponential form.

• Begin with \( e^{ \sum_{i=1}^{n} c_{i} \ln(x_{i}) } \).
• Exponentiate to derive the final result: \( \prod_{i=1}^{n} x_{i}^{c_{i}} \).

This transformation supports calculating the weighted geometric mean and verifies that the expression correctly simplifies to match theoretical expectations. Thus, the exponential function not only supports solving but confirms the logical progression of mathematical expressions in calculus.