Question

Show by examples that all expressions \(\left(1^{*}\right)\) are indeterminate.

Short answer

Expressions like \(1^{\infty}\) can lead to different outcomes, so they are indeterminate.

Step-by-step solution

Understand the concept of indeterminate forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions that do not have a clear limit. The indeterminate form related to this exercise is \(1^*\), which can result in different outcomes based on the context of the expressions within limits.

Set up a simple function that leads to 1^*

Consider the function \( y = (1+\frac{1}{x})^x \). As \(x \to \infty\), this expression approaches the number \(e\), showcasing that \(1^{\infty}\) can lead to a finite number different from 1.

Choose another function leading to 1^* context

Let's examine the function \(y = (1+\frac{1}{x})^{-x} \). As \(x \to \infty\), the expression evaluates to \(e^{-1}\) or \(\frac{1}{e}\), demonstrating an indeterminate form different from 1.

Evaluate yet another example of 1^*

Take the function \(y = \left(1+\frac{1}{x}\right)^{x^2} \). As \(x \to \infty\), this expression approximates \(0\), highlighting further the indeterminate nature of \(1^{\infty}\) leading to a very different value from the previous examples.

Key concepts

Limits in Calculus

Limits are fundamental in calculus, providing a way to understand the behavior of functions as they approach certain points or infinity. When we say we are "taking a limit," we mean predicting the value that a function approaches as the input changes. For example, if we have a function \( f(x) \), and we are interested in what happens as \( x \) approaches a particular number \( a \), we explore \( \lim_{x \to a} f(x) \).
A limit can be finite, infinite, or it might not exist. Calculating limits involves algebraic manipulation and applying limit laws, such as the quotient, product, and power rules. Limits help evaluate expressions where direct substitution is not possible due to indeterminate forms, such as \( \frac{0}{0} \) or \( 1^* \). Exploring limits can unlock a deeper understanding of continuity, derivatives, and integrals, making them indispensable in calculus.

Exponential Functions

Exponential functions have the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the exponent. They are essential in modeling growth and decay, featuring applications in science, finance, and mathematics. A key feature of exponential functions is their constant relative rate of growth or decay, meaning they increase or decrease by a fixed percentage over equal intervals.
In calculus, the natural exponential function \( e^x \) is particularly important. The number \( e \) is approximately 2.71828 and serves as the base of natural logarithms. It arises naturally in the context of rates of change and compounding processes.
Exponential functions exhibit unique properties:

• \( e^x \) is its own derivative and integral.
• \( e^0 = 1 \), establishing a useful point of reference.
• Exponential functions can approach limits and help understand indeterminate forms like \( 1^{\infty} \).

Their graphical representation shows rapid growth or decay, depending on whether the exponent is positive or negative.

Indeterminate Form Examples

Indeterminate forms occur when evaluating the limit of a function leads to an expression that does not initially suggest a particular value. The most common examples include \( \frac{0}{0} \), \( 0 \times \infty \), \( \infty - \infty \), and \( 1^{\infty} \). Each has unique characteristics, requiring special techniques to resolve, such as L'Hôpital's Rule, algebraic simplification, or using series expansions.
In the context of \( 1^{\infty} \), this form is particularly intriguing because it suggests multiplying 1 by itself an infinite number of times, which implies 1. However, the specific function involved might lead to various results when approaching the limit, as shown by our examples:

• \( y = (1 + \frac{1}{x})^x \to e \) as \( x \to \infty \), showcasing a convergence to Euler's number \( e \).
• \( y = (1 + \frac{1}{x})^{-x} \to \frac{1}{e} \), which changes the limiting value simply by altering the exponent's sign.
• \( y = (1+\frac{1}{x})^{x^2} \to 0 \), indicating how squared exponents alter the limit outcome entirely.

These examples showcase the flexibility and complexity of indeterminate forms, illustrating that context truly matters in the realm of limits and calculus.